6TH INTERNATIONAL CONFERENCE ON UNIFORM
DISTRIBUTION THEORY - UDT2018
October 1st – 5th 2018
CIRM, Marseille, FRANCE

Abstracts - BOOKLET
Radhakrishnan Nair - 60

WELCOME to UDT 2018https://www.cirm-math.com/
https://conferences.cirm-math.fr/1860.html
Following
UDT2008 at CIRM Marseilles,
UDT2010 in Strobl am Wolfgangsee, Austria
UDT2012 in Smolenice, Slovak Acad. of Sciences, Slovakia,
UDT2014 in Ostravice, Czech RepublicandUDT2016 in Sopron, Hungary,
we are honoured to welcome all our participants to CIRM for this, the latest in our series ofconferences. At UDT 2018 15 countries are represented with a total of 47 talks. The themes of theconference can be found on the homepage of the Journal UDT. We observe that at this meeting andelsewhere these themes are becoming more popular, and undergoing constant evolution, as thecommunity involved in its development grows. Sadly, a good number of colleagues were not able tocome due to the unfortunate timing of the meeting. This was an unhappy consequence of the
popularity of this venue.
Some statistics on the geographic origin of the participants and the "gender gap" are given below.
The topics of UDT are now very popular in Number Theory and its interactions with other domains,
as is evidenced by the large proportion of young researchers, amongst the perhaps three generations
that are involved in this meeting. We are proud to be the organisers of the present meeting and wish
you a pleasant and productive conference. We hope that UDT2018, your stay in Marseilles and at
CIRM will prove worthwhile.
UDT Journal
https://math.boku.ac.at/udt/

We also take the opportunity of this meeting to celebrate the 60th anniversary ofRadhakrishnan Nairby a text written by his colleague and friend Prof. Jaroslav Hančl,
for his numerous contributions tothis part of Number Theory and Dynamics he has honoured during all his life.
This text is available in the booklet below.
Scientific Committee UDT 2018
Shigeki Akiyama, University of Tsukuba
Arturas Dubickas, University of Vilnius
Christian Mauduit, Aix-Marseilles University
Radhakrishnan Nair, University of Liverpool
Oto Strauch, President UDT, Academy of Sciences at Bratislava
Jean-Louis Verger-Gaugry, CNRS, University Savoie Mont Blanc

Organizing Committee UDT 2018 :Oleg Karpenkov, University of Liverpool
Radhakrishnan Nair, University of Liverpool
Jean-Louis Verger-Gaugry, CNRS, University Savoie Mont Blanc
Secretariat UDT 2018:
Mrs Laure Bastide
LAMA, CNRS UMR 5127, University Savoie Mont Blanc
Bâtiment Le Chablais, campus scientifique,
73376 - Le Bourget-du-Lac, France
laure.bastide@univ-smb.fr
Tel: +33 (0)4 79 75 87 17
https://www.lama.univ-savoie.fr/
CNRS:
Mrs Emilie Morales
Service Financier et Comptable
Pôle Ressources Financières
CNRS Délégation Alpes
25, rue des Martyrs - BP 166
38042 Grenoble cedex 9, France
www.cnrs.fr/alpes
CIRM :Mrs Olivia Barbarroux,
CIRM - Luminy
Centre International de Rencontres Mathématiques
163 avenue de Luminy, Case 916
13288 Marseille cedex 9, France
Tel: +33 (0)4 91 83 30 00
https://conferences.cirm-math.fr/

CIRM : UDT 2018 , 6th International Conference on Uniform Distribution Theory.
Week October 1 - 5 2018
MondayTuesday

WednesdayThursdayFriday9.00-9.25

WelcomeLukas Spiegelhofer

Georges Grekos

Alexander Veselov

Michael Drmota

9.35-10.00

Andras Sarkozy

Arne Winterhof

Christine McMeekin

Ian Short

Bruno Martin

10.10-10.35

Imre Katai

Clemens Müllner

Rita Giuliano

Buket Eren

Mikhail Gabdullin

10.40-11.10

Coffee break

Coffee break

Coffee break

Coffee break

Coffee break

11.10-11.35

Olivier Ramaré

Anne Bertrand-Mathis

Thomas Stoll

Friedrich
Pillichshammer
Agamemnon
Zafeiropoulos

Mohammed Amri

11.45-12.10

Maria Avdeeva/
Mariia Monina
Iskander Aliev

12.30-14.00

Repas ­ lunch

Repas ­ lunch

Repas ­ lunch

Attila Pethoe

Repas – lunch

Alexander Kalmynin

Repas ­ lunch

UDT Committee
meeting14.30-14.55

Robert Tichy

Christoph
Aistleitner''Dmitry Gayfulin

15.05-15.30

Joël Rivat

Igor Pritsker''

Istvan Berkes

15.40-16.05

Arturas Dubickas

Dmitry Bilyk''

16.10-16.40

Coffee break

Coffee break

''

Photo
Coffee break

16.40-17.05

Alexey Ustinov

Oleg N. German''

Katalin Gyarmati

17.15-17.40

Joerg Thuswaldner

A.A. Illarionov''

Cathy Swaenepoel

17.50-18.15

Isabel Pirsic

Lisa Kaltenböck''

Vsevolod Lev

18.25-18.50

Jakub Konieczny

Sigrid Grepstad''

Iurii Shteinikov

Laszlo Merai''

Alexander Bufetov

DinnerBouillabaisse
Gala Dinner

19.00-19.25
19.30

DinnerDinnerJohn Blackman

closing

Leaving on Friday 5th September : keys to be brought back to the welcome desk
before 9.00 am.
Sessions on continued fractions

List of Chairwomen/Chairmen
SessionMonday9.30-10.40
11.10-12.20

Radhakrishnan Nair
Katalin Gyarmati

14.30-16.10
16.40-18.50

Cathy Swaenepoel
Alexander Veselov

9.00-10.40
11.10-12.20

Bruno Martin
Oleg Karpenkov

14.30-16.10
16.40-19.30

Attila Pethoe
Joerg Thuswaldner

Wednesday

9.00-10.40
11.10-12.20

Thomas Stoll
Arturas Dubickas

Thursday9.00-10.40
11.10-12.20

Alexey Ustinov
Robert Tichy

14.30-16.10
16.40-19.30

Laszlo MeraiJoël Rivat

9.00-10.40
11.10-12.20

Rita Giuliano
Olivier Ramaré

14.30-15.30

Jean-Louis Verger-Gaugry

Tuesday

FridayList of Speakers
Christoph Aistleitner (TU Graz) Pair correlations and equidistribution
Iskander Aliev (Cardiff University), Martin Henk and Timm Oertel On Distances to Lattice Points in Knapsack Polyhedra
Mohammed Amin Amri (Université Mohammed 1er) Uniform distribution of modular signs
Mariia Avdeeva (Institute of Applied Mathematics, Khabarovsk Division), Mariia Monina Basic properties of three-dimensional continued fractions
István Berkes (Graz University), Bence Borda Random walks on the circle and Diophantine approximation
Anne Bertrand-Mathis (Université de Poitiers) Expansions in negative base, distribution modulo one and fractales de Rauzy
Dmitriy Bilyk (University of Minnesota) Discrepancy and energy optimization on the sphereJohn Blackman (Durham University)
Integer multiplication of continued fractions via geometric methods
Alexander Bufetov (CNRS – Institut de Mathématiques de Marseille), Boris Solomyak A spectral cocycle for substitution dynamical systems
Michael Drmota (TU Vienna) The sum-of-digits function, primes and uniform distribution modulo 1
Artūras Dubickas (Vilnius University), Min Sha On the distance to the nearest square-free polynomial
Buket Eren (University of Galatasaray) , Muhammed Uludağ On the Markov equation and the outer automorphism of PGL(2, Z)
Mikhail Gabdullin (Lomonosov Moscow State University, N.N. Krasovskii Institute ofMathematics and Mechanics, Yekaterinburg)
On stochasticity parameter of quadratic residues
Dmitry Gayfulin (IITP of RAS), N. Shulga and I. Kan Minkowski question-mark function: fixed points and the derivative
Oleg German (Moscow State University) Multidimensional continued fractions and Diophantine exponents of lattices
Rita Giuliano (Università di Pisa) Rényi α-dimension of random variables with generalized Cantor distribution and Hausdorff
dimension of generalized Cantor sets
Georges Grekos (Université Jean-Monnet), Alain Faisant, Ram Krishna Pandey and Sai Teja Somu Sets with fairly distributed sumsets
Sigrid Grepstad (Johannes Kepler University, Linz) Asymptotic behaviour of the Sudler product of sines and a conjecture of Lubinsky
Katalin Gyarmati (Eötvös Loránd University, Budapest) On the cross-combined measure of families of binary lattices and sequences
Andrei Illarionov (Institute of Applied Mathematics, Khabarovsk Division) Statistical properties of Klein polyhedra
Alexander Kalmynin (National Research University, HSE) Large values of short character sums
Lisa Kaltenböck (Johannes Kepler University Linz), Gerhard Larcher On Bounded Remainder Sets and Strongly Non-Bounded Remainder Sets for Sequences({a_n α})_{n≥1}Imre Kátai (Eötvös Loránd University), Jean-Marie De Koninck Construction of normal numbers
Jakub Konieczny (Hebrew University of Jerusalem),T. Eisner Automatic and q-multiplicative sequences through the lens of higher order Fourier analysis

Vsevolod Lev (The University of Haifa at Oranim), Andras Biro Uncertainty in Finite Affine Planes
Bruno Martin (University Littoral Côte d Opale, Calais), Stéphane Jaffard Multifractal behaviour of the Brjuno function
Christine McMeekin (Cornell University) A density of ramified primes
László Mérai (Johannes Kepler University Linz) Distribution of short subsequences of the inversive generator
Mariia Monina (Institute of Applied Mathematics, Khabarovsk Division), Mariia Avdeeva Basic properties of three-dimensional continued fractions
Clemens Müllner (Université Claude Bernard Lyon 1) The Rudin-Shapiro sequence and similar sequences are normal along squares
Attila Pethő (University of Debrecen) Variations on a theme of K. Mahler
Friedrich Pillichshammer (Johannes Kepler University, Linz), Aicke Hinrichs and Shu Tezuka Tractability properties of the weighted star discrepancy
Isabel Pirsic (Johannes Kepler University, Linz), Roswitha Hofer An extension of the digital method based on b-adic integers
Igor Pritsker (Oklahoma State University) Uniform distribution for zeros of random polynomials
Olivier Ramaré (CNRS, Aix-Marseille Université) Discrepancy estimates for generalized polynomialsJoël Rivat (Aix-Marseilles University), Michael Drmota and Christian Mauduit On the digits of primes and squares
András Sárközy (Eötvös Loránd University, Budapest) Quasi-random graphs and pseudo-random binary sequences
Ian Short (Open University, Milton Keynes) The Farey graph, continued fractions and SL_2 -tilings
Iurii Shteinikov (Steklov Mathematical Institute and SRISA) On exponential sums and equations over multiplicative subgroups in finite fieldLukas Spiegelhofer (TU Wien) The level of distribution of the Thue-Morse sequence
Thomas Stoll (University of Lorraine, Nancy), R. de la Bretèche and G. Tenenbaum The sum of digits in two different bases
Cathy Swaenepoel (Aix-Marseille Université) Digital questions in finite fields
Jörg Thuswaldner (Montanuniversität Leoben) Discrepancy bounds for β-adic Halton sequences
Robert Tichy (TU Graz) Normality and Randomness
Alexey Ustinov (Russian Academy of Sciences, Khabarovsk) An elementary approach to Somos-4 sequences
Alexander Veselov (Loughborough University) Growth and Geometry in SL_2(Z) dynamics
Arne Winterhof (Austrian Academy of Sciences), Z. Sun On the maximum order complexity of subsequences of the Thue-Morse and Rudin-Shapiro
sequence along squares
Agamemnon Zafeiropoulos (Technical University of Graz), N. Technau Metric discrepancy with respect to fractal measures
The abstract « On subgraphs of random Cayley sum graphs », by S. V. Konyagin and I. D. Shkredov, has been added to
the list of abstracts.

Kit Nair — 60

Radhakrishnan Nair – Kit to everyone who knows him, was born in Tellicherry Kerala, South India to a Malayali mother, Anandavalli Nair – amathematics teacher and a Sri Lankan father Kothanath Balakrishnan Nair.
He grew up in Nigeria and Zambia in their early years of independence, where
his father worked as a Water Engineer. His final schooling years were in Ootyin Tamil Nadu, India, and when his parents seperated, in England, where hismother made her home.
He was an undergraduate and masters student in Imperial College, London. In his final years there, he took courses on, Uniform Distribution, Ergodic Theory, the Circle Method, Irregularities of Distributions, Harmonic
Analysis, Riemann Surfaces, the Riemann Zeta Function and Abstract Algebra, all of which influenced his subsequent work.
His initial intention was to study Analytic Number Theory at Imperial
but moved to Warwick to do his PhD in Ergodic Theory under William Parry– following available opportunities. He has retained an interest in arithmetic
questions all his professional life usually from a dynamical perspective.
On completion of his PhD in 1985, he was appointed as Research Assistant
to Mark Pollicott in Edinburgh, spent a year in Cambridge 1988-89, and was
appointed a lecturer in Liverpool in June 1989, where he has remained eversince. He was a Visiting Assistant Professor 1989–90 in Missouri and has
paid extended visits to Brest, Strasbourg and Ber Shiva in Israel as a British
Council Visiting Fellow. He also has a long mathematical association with
Ostrava, where he has jointly supervised students with Jaroslav Hančl.
Kit’s first mathematical interest was special arithmetic averages. He addressed a number of questions raised by R. C. Baker. His good fortune wasthat U. Krengel’s masterful book on ergodic averages appeared as he started
his PhD. This stimulated him to look at Baker’s questions as questions about
ergodic averages. This has largely led to complete resolution of these questions mostly by Kit.
Of particular note is the strong uniform distribution problem. In 1923 A.
Khinchinasked if given any B ⊂ [0, 1) of positive Lebesgue measure we havePN1χn=1 B ({nx}) → |B| almost everywhere in x with respect to LebesgueNmeasure. This was widely believed, but disproved by J. M. Marstrand in21970, who provided a Gδ counter example B. Marstrand also showed that if
a sequence of integers (mk )k≥1 is generated by finitely many distinct primes
R1
P
and f ∈ L∞ ([0, 1)) then N1 N
k=1 f ({mk x}) → 0 f (t)dt a.e. in x. We callsequences like this L∗∞ sequences.
In 1976 Baker asked if L∞ could be replaced by L1 . Kit showed this as afirst year PhD student, though it was published much later as an academic
career didn’t seem in prospect in the difficult 1980’s. This was his first
notable result. More recently, following work of G. Kozma, A. Quas and M.
Wierdl, he, with K. Chen, showed that a semigroup of integers is L∗1 if it
is contained in a finitely generated semigroup of integers. Otherwise they
showed there is f = χB with B being Gδ for which this convergence fails.
This brings a long chapter to a close.
Another prominent early result of his, developing earlier work of J. Bourgain and M. Wierdl was that if (X, β, µ, T ) is a dynamical system, φ(x)is a non-constant polynomial mapping the natural numbers to themselves,(pn )n≥1 is the sequence or primes, and f ∈ Lp (X, β, µ) for p > 1 thenPφ(pn )
limN →∞ N1 Nx) exists almost everywhere. Notice that the denn=1 f (Tsity version of Dirichlet’s theorem on arithmentic progressions is a special
case of this result, showing that it is in fact an ergodic theorem. Laterhe showed that such an ergodic theorem is true for a very broad class ofHartman uniformly distributed sequences of integers. This family of results
has broad application. Kit has written many papers refining the KhinchinDoeblin theory of continued fractions to subsequences, opening many newlines of investigation in the process.
A sequence of integers for which the ergodic theorem is true is called
good universal. A dynamical system is called uniquely ergodic if there isexacly one measure µ such that (X, β, T, µ) is ergodic. Recently, extending
work of J. Oxtoby, with his students
that (X, β, T, µ) is uniquely
P he showed
anergodic if and only if limN →∞ N1 N
f
(T
x)
exist
uniformly in x when X
n=1
is compact, and f ∈ C(X) for sequence if (an )n≥1 that are Hartman uniformly distributed and good universal. This is then used to show the van der
Corput sequence and its generalisations (vβ (an ))n≥1 are uniform distributedfor Hartman uniformly distributed good universal sequences (an )n≥1 . Thisbuilds on work of R Tichy and his co-authors.
Let P denote a polynomial with some coefficient other than P (0) irrational. H. Weyl showed (P (an ))n≥1 is u.d. mod 1 if an = n. I. M. Vinogradov
and G. Rhin dealt with the case an = pn . Kit extended this to all Hartmam

3

uniformly distributed good universal (an )≥1 .
A sequence K = (bn )≥1 is called intersective if, given any set of integers
S of positive density there exist x, y ∈ S such that x − y ∈ K. Developing
earlier work of A. Sarkozy and H. Furstenberg on squares for instance Kit
showed (a2n )n≥1 is intersective if (an )n≥1 Hartman uniform distributed andgood universal. In related work he classified P where (P (pn ))n≥1 is intersective and in fact showed that Hartman uniform distributed on any metriablegroup, is intersective.
This is a sample of his work.
Kit is still research-active and expects to remain so as long as he is fit andable. He is proudest about what his students have achieved, however. He has
many good friends throughout the profession. Away from work, he spends
time with his family, his dog and two cats, and other creatures uninterested
in mathematics.

September 2018
Prof. Jaroslav Hančl6th International Conference on Uniform Distribution Theory
PARTICIPANTS – EMAILSFAMILY NAME
Aistleitner
AkiyamaAliev
AmmousAmri
ArnouxAvdeeva
Balazard
Ben Mahfoudh
BerkesBertrand Mathis
Beyaoui
Bilyk
Blackman
Bufetov
Cassaigne
DrmotaDubickas
Eren
Ferenczi
Frid
Gabdullin
Gayfulin
GermanGiuliano
GrekosGrepstadGyarmati
Habsieger

FIRST NAME
Christoph
ShigekiIskanderBasma
Mohammed Amin
PierreMariiaMichelLobna
IstvanAnne
Salma
DmitriyJohn
Alexander
JulienMichaelArtüras
Buket
SébastienAnna
MikhailDmitryOleg
Rita
Georges
SigridKatalinLaurent

EMAIL
aistleitner@math.tugraz.atakiyama@math.tsukuba.ac.jp
alievi@cf.ac.uk
ammous.basma@hotmail.fr
amri.amine.mohammed@gmail.com
pierre@pierrearnoux.fr
avmariya@yandex.rubalazard@math.cnrs.fr
lobna.ben.mahfoudh2@gmail.com
berkes@tugraz.at
anne.bertrand@math.univ-poitiers.fr
bayaouisalma1990@gmail.com
bilyk.dmitriy@gmail.com
john.blackman@durham.ac.uk
alexander.bufetov@univ-amu.fr
julien.cassaigne@math.cnrs.fr
michael.drmota@tuwien.ac.atarturas.dubickas@mif.vu.lt
bktern@hotmail.com
ssferenczi@gmail.com
anna.e.frid@gmail.com
gabdullin.mikhail@yandex.ru
gamak.57.msk@gmail.com
german.oleg@gmail.com
rita.giuliano@unipi.it
grekos@univ-st-etienne.fr
sgrepstad@gmail.com
gykati@cs.elte.hu
habsieger@math.univ-lyon1.fr

HbaibIllarionov
Kalmynin
Kaltenböck
Karpenkov
Kátai
Konieczny
Laurent
Lev
MarnatMartinMauduit
McMeekin
Mérai
MoninaMüllner
Nair
Nguema Ndong
Pethö
Pillichshammer
Pintz
PirsicPritsker
Ramaré
Rivat
Sárközy
Short
Shteinikov
Spiegelhofer
Steiner
Stoll
Swaenepoel
Thuswaldner
Tichy
Troubetzkoy
Ustinov
Verger-Gaugry
Veselov
WannesWinterhof
Zafeiropoulos
Zhuravleva

MohamedAndrei
Alexander
Lisa
Oleg
Imre
Jakub
MichelVsevolod
Antoine
Bruno
ChristianChristine
László
MariiaClemens
Radhakrishnan
Florent
AttilaFriedrich
Janos
IsabelIgor
OlivierJoëlAndrás
Ian
Iurii
Lukas
WolfgangThomas
Cathy
Jörg
RobertSerge
AlexeyJean-Louis
Alexander
Walid
Arne
Agamemnon
Victoria

mmmhbaib@gmail.com
illar_a@list.ru
alkalb1995cd@mail.ru
lisa.kaltenboeck@jku.at
karpenk@liv.ac.uk
katai@inf.elte.hu
jakub.konieczny@gmail.com
michel-julien.laurent@univ-amu.fr
seva@math.haifa.ac.il
antoine.marnat@york.ac.uk
Bruno.Martin@univ-littoral.fr
Mauduit@iml.univ-mrs.fr
cem255@cornell.edu
laszlo.merai@oeaw.ac.atmonina_dvggu@mail.ru
mullner@math.univ-lyon1.fr
nair@liv.ac.uk
florentnn@yahoo.fr
Petho.Attila@inf.unideb.hu
friedrich.pillichshammer@jku.at
pintz@renyi.hu
isabel.pirsic@jku.at
igor@math.okstate.edu
olivier.ramare@univ-amu.fr
joel.rivat@univ-amu.fr
sarkozy@cs.elte.hu
ian.short@open.ac.uk
yuriisht@yandex.rulukas.spiegelhofer@tuwien.ac.at
steiner@irif.fr
thomas.stoll@univ-lorraine.fr
cathy.swaenepoel@univ-amu.fr
jorgthus@gmail.com
tichy@tugraz.atserge.troubetzkoy@univ-amu.fr
ustinov.alexey@gmail.com
Jean-Louis.Verger-Gaugry@univ-smb.fr
A.P.Veselov@lboro.ac.uk
wannes_walid@ymail.com
arne.winterhof@oeaw.ac.at
zafeiropoulos@math.tugraz.at
victoria.zhuravleva@me.com

ABSTRACTSPair correlations and equidistribution
Christoph Aistleitner

TU Grazaistleitner@math.tugraz.at
Consider an infinite sequence in the unit interval. The pair correlation
statistic quantifies how many pairs of elements of the sequence (with index
up to a fixed value) are within a certain short distance of each other. Here“short” means a distance proportional to the average spacing between the
points. If the asymptotic distribution of pair correlations coincides with thatof a random sequence, we say that the pair correlation behavior is Poissonian.
Mathematical folklore presumed that having Poissonian pair correlations isa “finer” property than being equidistributed. In this talk we give a rigorous
proof that Poissonian pair correlations imply equidistribution. For the proof
we will use a blend of Fourier analysis and linear algebra.

5

On Distances to Lattice Points in Knapsack Polyhedra
Iskander Aliev1 , Martin Henk2 , and Timm Oertel3
1

Mathematics Institute, Cardiff University, UK
alievi@cardiff.ac.uk
2
Department of Mathematics, TU Berlin, Germany
henk@math.tu-berlin.de3
Mathematics Institute, Cardiff University, UK
oertelt@cardiff.ac.uk

Abstract. Given a ∈ Zn , b ∈ Z, a knapsack polyhedron P (a, b) is defined asT
P (a, b) = {x ∈ Rn
≥0 : a x = b} .

In this talk we will present some recent results on the ℓ∞ -distance from a vertex ofa knapsack polyhedron to its nearest feasible lattice point (referred to as a vertex
distance). We give a sharp upper bound for the vertex distance that only depends
on the ℓ∞ -norm of the vector a. In a randomised setting, we show that the vertex
distance for a “typical” knapsack polyhedron is drastically smaller than the vertex
distance that occurs in a worst case scenario.

6

Uniform distribution of modular signs
Mohammed Amin Amri
August 3, 2018

Abstract
Let k ≥ 2, N be natural numbers, and letXf=a(n)q nq := e2πiz

n≥1

be a normalised newenform of integral weight k for Γ0 (N ) with Dirichlet character ε (mod N ).
In this talk, we will consider the sign changes and the distribution of signs of some sub-families ofthe sequence (ℜe {a(n)e−iφ })n∈N , with φ ∈ [0, π). Using the now-proven Sato-Tate Conjecture
for modular forms we prove some equidistribuion results of the signs of these sequences.
Mohammed Amin Amri, ACSA Laboratory, Department of Mathematics,Faculty of Sciences, Mohammed First University,Oujda, Morocco.
E-mail address, Mohammed Amin Amri: amri.amine.mohammed@gmail.com

1

7

Basic properties of three-dimensional
continued fractions
Mariia Avdeeva, Mariia Monina

Pacific National University, RUSSIA
avmariya@yandex.ru, monina dvggu@mail.ru

Let the continued fraction representation of real α is a [t0 ; t1 , . . . , ti , . . . ],where t0 ∈ Z and ti ∈ Z+ (i = 1, 2, . . . ) are called the coefficients or termsof the continued fraction and pi /qi = [t0 ; t1 , . . . , ti ] (i = 1, 2, . . . ) are called
the convergens of the continued fraction.
In the continued fractions theory we know the Markov-Hurwitz theorem:∀i = 1, 2, . . .
1
pi−1 2pi 2
pi+1 2qi−1 , α −qi , α −qi+1 ≤ √ .
min α −
qi−1
qi
qi+1
5
The talk will be devoted to the analog of the theorem for three dimensional lattices.
This research is supported by the Russian Science Foundation (project
18-14-05001).

8

Random walks on the circle and Diophantine
approximation
István Berkes, Rényi Institute of Mathematics
Let (nk ) be an increasing sequence of positive integers and let α be an irrational number. The asymptotic behavior of the discrepancy of {nk α}1≤k≤N iswell known for nk = k and for lacunary nk , but otherwise no precise discrepancy
estimates are known. Thus it is natural to consider random nk and we investigate the simplest case when the gaps nk+1 − nk are independent and identically
distributed random variables. Then {nk α} is a random walk on the circle having
interesting and unusual properties. By Markov chain theory, the distribution of{nk α} converges to the uniform distribution on the circle, but the convergence
speed rn = supx |P ({nk α} < x) − x| is much slower than the exponential speed
for finite Markov chains and depends strongly on the Diophantine approximation properties of α. We give a precise Berry-Esseen type estimate for rn anddescribe the asymptotic properties of the discrepancy DN of {nk α}1≤k≤N andP
TN = N
k=1 f (nk α) for periodic measurable functions f . In particular, letting
γ denote the Diophantine rank of α, we show that the order of magnitude of
TN and DN is substantially different for γ ≤ 2 and γ > 2, a surprising criticalphenomenon in the model.
Joint results with Bence Borda.

Email: berkes.istvan@renyi.mta.hu

9Expansions in negative base, distribution
modulo one and fractales de Rauzy.
September 5, 2018
Anne Bertrand Mathis
Université de Poitiers (France)
anne.bertrand@math.univ-poitiers.fr
Abstract
Given a Pisot number with degree d we define three natural maps fromthe so-called −β shift into R and Rd and onto the d−dimensional torus ;these maps allows us to define a “Pavé de Rauzy” and its multiple tiling of
Rd , a fractale analoguous to Rauzy fractale and a multiple autoaffine tilingof Rd−1 with negative expansion value ; we also show that a number has
a normal expansion in base −β if and only if he has a normal expansion
in base +β.

Bibliography:[Aki] S. Akiyama. Self-affine tiling and Pisot numeration system.
Number Theory and its applications (Kyoto 1997) vol2 Devt.Math,Vol 2 p
7-17.
[Am] P. Ambroz, D. Dombek, Z. Makarova, E. Pelantova. Numbers withinteger expansion in the numeration system with negative base. Preprint.
[BS] V.Berthé, A. Siegel. Tilings associated with Beta-numeration and substitutions. Electronic Journal of Combinatorial Number Theory 5 (3) (2005).
[BM ] A. Bertrand-Mathis. Développements en base θ, répartition modulo
un de la suite (xθn )n≥0 , langages codés et θ-shifts. Bull.Soc.Math.France 114
27-323 (1986).
[BlH] F.Blanchard, G.Hansel. Systemes codés. Theoretical computer sciences 44 17-49 (1986)[DGS] M. Denker, C. Grillenberger, K. Sigmund. Ergodic theory on compact
spaces . Lect.Notes.Math.527 Springer Berlin1976.
[F L] C.Frougny, A.C. Lai. On negative bases. Proceeding of DLT 09 Lectures Notes in Computer Sciences 5583 (2OO9).
[IS] S.Ito, T.Sadahiro. βexpansions with negative bases. Integers 9 (2009)
A22 239-259.
[KN ] L.Kuipers, H. Niederreiter. Uniform distribution of sequences. Pure
ans Applied Mathematics Wiley -Intersciences New-York 1974.

1

10[L.S.] L.Liao,W. Steiner. Dynamical transformation of the negative betatransformation.Ergodic Th. Dynam Systems 32 (2012) 1673-1690.
[N.N 1] F. Nguéma Ndong. On the Lyndon Dynamical Systems. Adv.in
app.Math.78 (2016)1-26[N.N 2] F.NguémaNdong. Sur la discrimination des nombres normaux. Preprint.
[Ra] G.Rauzy. Nombres algébriques et substitutions. Bull.Soc. Math.
France110 (1982) n°2 147-178.
[Re] A. Rényi. Representations for real numbers and their ergodic properties,Acta Math.Acad.Sci.Hungar 8 477-493 (1957).
[S] N.Sidorov. Bijective and general arithmetic codings for Pisot toral automorphisms.J.Dynam. Control systems 7 (2001) n°4 447- 472.
[W a] P. Walters. Ergodic Theory, Introducting Lectures. Lecture notes in
Math. vol 458 (1975)Springer, Berlin.
[W e] H.Weyl. Uber die Gleichverteilung der Zahlen mod. Eins. Math.
Annalen 77 (1916) n°3 313-352.
Anne Bertrand-Mathis
Département de Mathématique
SP2MI
86962 Futuroscope cedex
bertrand@math.univ-poitiers.fr
.

2

11

Dmitriy Bilyk
University of Minnesota
dbilyk@math.umn.edu
Discrepancy and energy optimization on the sphere.
Abstract: Discrepancy and optimal energy are among the most standard ways to measure equidistribution of finite point sets. There
first explicit connection between certain instances of these notions (L2spherical cap discrepancy and sum of distances) was provided by the
Stolarsky invariance principal, which says that minimizing the former
is equivalent to maximizing the latter. This principle has a experienced renewed attention in the recent years with numerous new proofs,
different versions, extensions, and generalizations. We shall survey
different manifestations of this principle and some of their applications.We also explore measures and discrete point distributions which
minimize various energies on the sphere. Unlike classical examples,such as the Riesz energy, most of the potentials that we consider areminimized when points are orthogonal, thus minimizing energy imposes orthogonal-like structures (bases, frames etc). In particular, weconsider the so-called p-frame energy: in the well-known case p = 2
the minimizers are precisely tight frames (or, more generally, isotropic
measures) on the spheres, however, other values of p > 0 are muchless understood (except even integers). We provide minimizers in somespecific cases, as well as a number of candidates for other cases basedon extensive numerical computations. We also present partial progresson the conjecture of Fejes Toth, which states that the pairwise sum of
acute angles between n lines is maximized by the periodically repeatedorthonormal basis.

Integer Multiplication of Continued Fractions Via
Geometric Methods
John Blackman
Durham University, UK
john.blackman@durham.ac.uk
In recent years, the problem of deducing the continued fraction expansion of pα from thecontinued fraction of α has become a topic of increased interest. This is due in part to a reformulation of the p-adic Littlewood conjecture – an open problem in Diophantine approximation– which roughly states that the partial quotients of pk α become arbitrarily large as k tends toinfinity (for p a fixed prime).
In this talk, we will discuss how one can interpret multiplication of a continued fraction by nsome integer as a map between the Farey complex and the 1/n-scaled Farey complex. In turn,this allows us to interpret the integer multiplication of continued fractions as a replacement of
one triangulation of Γ0H with another triangulation. We will then discuss how closed curves
on Γ0H directly correspond to periodic continued fractions, and how this correspondence allows
us to deduce information about the divisibility of convergent denominators for α as well as thegrowth rate of partial quotients of pk α for α eventually periodic.

1

A SPECTRAL COCYCLE FOR SUBSTITUTION DYNAMICAL SYSTEMS (JOINT WITH
BORIS SOLOMYAK)
Alexander Bufetov (CNRS Institut de Mathématiques de Marseille,
alexander.bufetov@univ-amu.fr)

The talk, based on joint work with Boris Solomyak, is devoted to quantitative
spectral estimates for suspension flows over substitution dynamical systems. In thissituation, the spectrum is described by matrix analogues of Riesz products.We
introduce a new cocycle over the skew-product whose base is our system and whose
fibres are toral automorphisms. The Lyapunov exponent of the cocycle governs the
spectral measure.

12

The sum-of-digits function, primes anduniform distribution modulo 1
Michael Drmota

TU Wien, Austria
michael.drmota@tuwien.ac.at

There is an intimate relation between the distribution of the q-ary sumof-digits functions of primes sq (p), the uniform distribution of the sequence
αsq (p) mod 1 and on special instances of the Sarnak conjecture. This kindof work war pioneered by the ground-breaking work by Mauduit und Rivaton the Gelfond-problems.
The purpose of this talk is to give a survey of recent results into thisdirections that have been obtained together with Mauduit and Rivat andalso with Müllner and Spiegelhofer. In particular we show that the sequence
αsq1 (p) + βsq2 (p) mod 1 is uniformly distributed (for coprime q1 , q2 and irrational α, β) and that the sequence sZ (n) mod 2 satisfies the Sarnak conjecture, where sZ (n) denotes the Zeckendorf sum-of-digits function.

13

On the distance to the nearest square-free polynomial
Artūras Dubickas (Vilnius University)e-mail: arturas.dubickas@mif.vu.lt
We investigate a variant of Turán’s problem on the distance from an
integer polynomial in Z[x] to the nearest irreducible polynomial in Z[x],where ,,irreducible” is replaced by ,,square-free”.
In particular, we show that, for any polynomial f ∈ Z[x], there exist
infinitely many square-free polynomials g ∈ Z[x] such thatL(f − g) ≤ 2,
where L(h) denotes the sum of the absolute values of the coefficients ofh ∈ Z[x]. (We do not know if such a polynomial g exists if we require, in
addition, deg g ≤ deg f , but we can bound deg g in terms of d = deg f andL(f ).) On the other hand, we show that this inequality cannot be replacedbyL(f − g) ≤ 1.
For this, for each integer d ≥ 15 we construct infinitely many polynomialsf ∈ Z[x] of degree d such that neither f itself nor any f (x) ± xk , where k isa non-negative integer, is square-free. One example of degree 15 polynomialisf (x) =15552x15 + 5184x14 + 5616x13 + 8784x12 + 13908x11+ 13756x10 + 96413x9 − 18929x8 − 57229x7 + 6851x6+ 9435x5 − 932x4 − 346x3 + 36x2 .
Polynomials over prime fields and their distances to square-free polynomials are also considered.
These results are joint with Min Sha (Macquarie University, Sydney).

14

On the Markov equation and the outerautomorphism of PGL(2, Z)
Buket Eren (joint work with Muhammed Uludağ)Galatasaray University
bktern@hotmail.com
Abstract
The Markov numbers are the unions of the solutions (x, y, z) ∈ Z3+ tothe Markov equation x2 +y 2 +z 2 = 3xyz. These numbers arise in many different contexts. Our interest is in the set of continued fraction expansions ofMarkov quadratic irrationals arised from Markov numbers. However, there
is a fundamental involution of the real line called Jimm induced by the outerautomorphism of the extended modular group PGL(2, Z). Its action on thereal line, explicitly on continued fraction expansion, inspired us that this involution must play a role in Markov’s theory. The main goal of this talk is toshow the possible relation between Jimm involution and the Markov theory.
More precisely, we will first present a method to find directly the quadratic
form of the image of Markov irrationals under Jimm involution by using
the fact coming from geometry of Markov numbers. Then we will show ageneral form of the image of the subset of Markov quadratic irrationals in
relation with Fibonacci numbers.

ON STOCHASTICITY PARAMETER OF QUADRATICRESIDUES
Mikhail Gabdullin 1
Lomonosov Moscow State University
N.N. Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg
Let A be an arbitrary subset of residue ring Zm . We can write
A = {ai }ti=1 , where 0 6 a1 < a2 < . . . at < m, and set at+1 := a1 + n (thus
we think of Zm as of a circle). For studying the degree of randomness ofpoints of A Arnold [1], §9, define a stochasticity parameter
S(A) :=

tX(ai+1 − ai )2 .

i=1

Too small or too large values of S(A) testify to nonrandom behaviour of
A: the points of A either “repel” or “attract” each other (S(A) is minimal
when the points are distributed equidistantly, and is maximal when all the
points accumulate in the same place).
We investigate the quantity S(R), where R is the set of quadratic
residues modulo m. Let S(k), k ∈ N, be a mean value of S over all kelements sets of Zm . A special case of a result of Konyagin, Garaev andMalykhin [2] is the following.
Theorem A. Let m = p be a prime number. Then
S(R) = S(|R′ |)(1 + o(1)),

p→∞

We present the following results.
Theorem 1. Let a ∈ N be a fixed positive integer and m = ap1 . . . pt ,where p1 < . . . < pt are primes and p1 is sufficiently large depending on aand t. Then
S(R) = S(|R|)(1 + o(1)), t → ∞.
Theorem 2. We have
S(R)
S(R)
< 1 < lim.
m→∞ S(|R|)
m→∞ S(|R|)
lim

1

The work is supported by the grant from Russian Science Foundation (Project 14-11-00702).

1References[1] V.I.Arnold, Euler Groups and Arithmetics of Geometric Progressions(MTsNMO, Moscow, 2003) [in Russian].
[2] M.Z.Garaev, S.V.Konyagin, Yu.V.Malykhin, “Asymptotics for thesum of powers of distances between power residues modulo a prime”,
Proceedings of the Steklov Institute of Mathematics, 276 (1), 2012,
pp 77-89.
gabdullin.mikhail@yandex.ru

2

15▼✐♥❦♦✇s❦✐ q✉❡st✐♦♥✲♠❛r❦ ❢✉♥❝t✐♦♥✿ ✜①❡❞ ♣♦✐♥ts ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡❉♠✐tr② ●❛②❢✉❧✐♥■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠s✱ ▼♦s❝♦✇✱ ❘✉ss✐❛❚❤❡ ▼✐♥❦♦✇s❦✐ q✉❡st✐♦♥✲♠❛r❦ ❢✉♥❝t✐♦♥ ?(x) ✐s ❛ ❝♦♥t✐♥✐♦✉s ❛♥❞ str✐❝t❧②✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ♠❛♣s t❤❡ [0, 1] ✐♥t❡r✈❛❧ ♦♥t♦ ✐ts❡❧❢✳ ■❢ [0; a1 , a2 , . . . , an , . . .]✐s t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r x✱ t❤❡♥?(x) =∞
X
k=1(−1)k+1
2a1 +...+ak −1✭✶✮■❢ x ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✱ t❤❡♥ ✭✶✮ ✐s r❡♣❧❛❝❡❞ ❜② ✜♥✐t❡ s✉♠✳ ❖♥❡ ❝❛♥ ❡❛s✐❧②s❡❡ t❤❛t t❤❡ ❡q✉❛t✐♦♥?(x) = x✭✷✮❤❛s ❛t ❧❡❛st ✺ s♦❧✉t✐♦♥s✱ ❛ ❢♦❧❦❧♦r❡ ❝♦♥❥❡❝t✉r❡ st❛t❡s t❤❛t t❤❡r❡ ❛r❡ ❡①❛❝t❧②✺ ✜①❡❞ ♣♦✐♥ts ♦❢ ?(x)✳ ■ ✇✐❧❧ t❡❧❧ ❛❜♦✉t s♦♠❡ ♥❡✇ r❡s✉❧ts ♦♥ ❉✐♦♣❤❛♥t✐♥❡♣r♦♣❡rt✐❡s ♦❢ ✜①❡❞ ♣♦✐♥ts ♦❢ ?(x) ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✳ ❙❤✉❧❣❛✮ ❛♥❞ ♣r♦♣❡rt✐❡s♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ■✳ ❑❛♥✮✳

✶

16

Multidimensional continued fractions andDiophantine exponents of lattices
Oleg N. German
Moscow State University
The talk is devoted to discussing multidimensional continued fractions and connected problems concerning Diophantine exponents of lattices.
Given a full rank lattice Λ in Rd we define its Diophantine exponent asonω(Λ) = sup γ ∈ R Π(x) < |x|−γ admits ∞ solutions in x ∈ Λ ,where Π(x) = |x1 · . . . · xd |1/d if x = (x1 , . . . , xd ). This quantity generalizes to the caseof lattices the concept of irrationality measure of a real number. It is well known thatthe measure of irrationality of θ can be expressed in terms of the growth of partial
quotients of θ. Namely,ln(an+1 )µ(θ) = 2 + lim sup,ln(qn )n→∞where an are the partial quotients of θ, and qn are the denominators of its convergents.
We shall discuss possible multidimensional generalizations of this connection, considering Klein polyhedra as a multidimensional generalization of continued fractions.

1

17Rényi α−dimension of random variables with generalized Cantor
distribution and Hausdorff dimension of generalized Cantor sets
Rita Giuliano∗

Abstract
We discuss the notion of generalized Cantor set and generalized Cantor distribution; we introduce the notion of Rényi α−dimension and prove that the Rényi lower α−dimension of arandom variable with generalized Cantor distribution coincides with the Hausdorff dimension ofits support (generalized Cantor set). Under some assumptions, we also prove the holderianityof generalized Cantor distribution functions.

∗
Address: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa, Italy. e-mail:rita.giuliano@unipi.it

1

19

Sets with fairly distributed sumsets
Georges Grekos

Université de Saint-Étienne, 23 rue du Dr Paul Michelon,42023 ST ETIENNE Cedex 2, France
grekos@univ-st-etienne.fr
https://perso.univ-st-etienne.fr/grekos/

This is a talk on a common work with Alain Faisant (Saint-Étienne),
Ram Krishna Pandey (Roorkey) and Sai Teja Somu (Mumbai). Startingfrom a problem in Additive Number Theory (given α ∈ [0, 1], find A ⊆ N :={0, 1, 2, . . . } such that d(A + A) = α, where the “asymptotic density” d isdefined as d(X) := limn→+∞ n−1 |X ∩ [1, n]|), we found a manner to construct
sets A of nonnegative integers such that each sumset jA, 1 ≤ j ≤ k, is fairly
distributed in N. A sumset jA is defined by jA := {x1 + · · · + xj ; xi ∈ A, 1 ≤i ≤ j}. By “fair” we mean that the sumset has asymptotic density. Moreoverthe densities of the sumsets are regularly distributed in [0,1].

20

Asymptotic behaviour of the Sudler product ofsines and a conjecture of Lubinsky.
Sigrid Grepstad
September 13, 2018
We study the asymptotic behaviour of the sequence of sine products
Pn (α) =nY|2 sin πrα|r=1

for real quadratic irrationals α. In particular, we study the subsequence Pqn ,where qn is the nth best approximation denominator of α, and show that
(Pqn )n≥1 converges to a periodic sequence whose period equals that of thecontinued fraction expansion of α. This verifies a conjecture recently posedby Mestel and Verschueren in [1]. Finally, we discuss a conjecture of Lubinsky[2] stating thatlim
inf Pn (α) = 0,n→∞
and argue that this new information on the asymptotic behaviour of Pn (α)
suggests otherwise.

References[1] P. Verschueren and B. Mestel, Growth of the Sudler product of sines at
the golden rotation number, J. Math. Anal. Appl. 433 (2016), 200–226.
[2] D. S Lubinsky, The size of (q; q)n for q on the unit circle, J. Number
Theory 76 (1999), 217–247.

1

21

On the cross-combined measure of families ofbinary lattices and sequences
Katalin Gyarmati
ELTE Eötvös Loránd University,
Institute of Mathematics,
Department of Algebra and Number Theory
gykati@cs.elte.hu

The cross-combined measure (which is a natural extension of crosscorrelation measure) is introduced and important constructions of large families of binary lattices with nearly optimal cross-combined measures arepresented. These results are important in the study of large families ofpseudorandom binary lattices but they are also strongly related to the onedimensional case: An easy method is showed obtaining strong constructionsof families of binary sequences with nearly otimal cross-correlation measures
based on the previous constructions of lattices. The important feature of thisresult is that so far there exists only one type of constructions of very large
families of binary sequences with small cross-correlation measure, and this
only type of constructions was based on one-variable irreducible polynomials.
Since it is very complicated to construct one-variable irreducible polynomials over Fp , it beame necessary to show other types of constructions where
the generation of sequences are much faster. Using binary lattices based on
two-variable irreducible polynomials this problem can be avoided, howevera slightly weaker upper bound is obtained for the cross-correlation measurethan in the original construction. (But, contrary to one-variable polynomials,
using Schöneman-Eisenstein criteria it is very easy to construct two-variable
irreducible polynomials over Fp .)1

23

Statistical properties of Klein polyhedra
A.A. Illarionov(Pacific National University, Khabarovsk, Russia),
e-mail: illar a@list.ru
Let Γ be an s-dimensional lattice. For each θ = (θ1 , . . . , θs ), θi = ±1,
define the Klein polyhedron Kθ (Γ) as the covex hull of nonzero lattice points
contained in the s-dimensional polyhedral conex ∈ Rs : xi θi ≥ 0, i = 1, s .
Klein’s construction was motivated the following considerations. For eachα ∈ (0, 1) define the lattice Γα of points γ = (Q, α · Q − P ) with Q, P ∈ Z.
By the classical Lagrange approximation theorem, the set of vertices of theKlein polyhedra for the lattice Γα consists of the points±(0, 1),±(Qn , αQn − Pn ),n = 0, 1, . . . ,where Pn /Qn — is the nth convergent for α. Futhemore, leta = (Qn−1 , αQn−1 − Pn−1 )ã = (Qn+1 , αQn+1 − Pn+1 )be two adjacent vertices, that is, the segment [a, ã] is an edge of a Klein
polyhedron for Γα . Then
qn+1 (α) = #(Γ ∩ (a, ã]).

(1)Here #X denotes the cardinality of X.
We will discuss some statistical properties of multydimensional Klein
polyhedra (see [1–4]).
1. O. Karpenkov, Geometry of continued fractions, Algorithms Comput.
Math., vol. 26, Springer, Berlin-Heidelberg, 2013.
2. A. A. Illarionov, ”On the statistical properties of Klein polyhedra inthree-dimensional lattices”, Sbornik: Mathematics, 204:6 (2013).
3. A. A. Illarionov, ”Some properties of three-dimensional Klein polyhedra”, Sbornik: Mathematics, 206:4 (2015), 35–66 (2015).
4. A. A. Illarionov, ”Distribution of facets of higher-dimensional Klein
polyhedra”, Sbornik: Mathematics, 209:1 (2018).

24❆❧❡①❛♥❞❡r ❑❛❧♠②♥✐♥▲❛r❣❡ ✈❛❧✉❡s ♦❢ s❤♦rt ❝❤❛r❛❝t❡r s✉♠s■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t s✉♠s ♦❢ ✈❛❧✉❡s ♦❢ ♥♦♥✲♣r✐♥❝✐♣❛❧ ❉✐r✐❝❤❧❡t ❝❤❛r✲❛❝t❡rs ♦✈❡r ❧❛r❣❡ ❡♥♦✉❣❤ ✐♥t❡r✈❛❧s ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ♥♦♥✲tr✐✈✐❛❧❧②✳ ◆❛♠❡❧②✱ ❢♦r ❛♥②
ε > 0 t❤❡r❡ ❡①✐sts c(ε) > 0 s✉❝❤ t❤❛t ❛♥② ❝❤❛r❛❝t❡r s✉♠ ♦❢ ❧❡♥❣t❤ N > p1/4+ε ✐sO(N p−c(ε) )✳ ❍♦✇❡✈❡r✱ ✐t t✉r♥s ♦✉t t❤❛t ❢♦r ❛♥② A > 0 t❤❡r❡ ❡①✐st ✐♥✜♥✐t❡❧② ♠❛♥②
♣r✐♠❡s p ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ♥♦ ♥♦♥✲tr✐✈✐❛❧ ❜♦✉♥❞ ❢♦r t❤❡ s✉♠ ♦❢ ✈❛❧✉❡s ♦❢ q✉❛❞r❛t✐❝❝❤❛r❛❝t❡r ♠♦❞✉❧♦ p ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ (log p)A ✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss t❤❡ ♣r♦♦❢♦❢ t❤✐s r❡s✉❧t✱ ✇❤✐❝❤ r❡❧✐❡s ♦♥ s♦♠❡ ❝❧❛ss✐❝❛❧ ❢❛❝ts ❛❜♦✉t ❙✐❡❣❡❧ ③❡r♦s ❛♥❞ s♠♦♦t❤
♥✉♠❜❡rs✱ ❛♥❞ ✐ts ♣♦ss✐❜❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s✳
❆❜str❛❝t✳

❝❆❧❡①❛♥❞❡r ❑❛❧♠②♥✐♥✱

✷✵✶✽

25

On Bounded Remainder Sets andStrongly Non-Bounded Remainder Sets for Sequences({anα})n≥1
Lisa Kaltenböck∗, Gerhard Larcher†
Abstract: Let (xn )n≥1 be an arbitrary sequence in [0, 1). An interval [a, b) ⊆ [0, 1) is calledbounded remainder set (BRS) for (xn )n≥1 if|#{1 ≤ n ≤ N : xn ∈ [a, b)} − (b − a)N | ≤ c,for all natural numbers N with a constant c independent of N . We give some results on theexistence of BRS for sequences of the form ({an α})n≥1 , where (an )n≥1 - in most cases - is agiven sequence of distinct integers. Further we introduce the concept of strongly non-boundedremainder sets (S-NBRS) and we show for a very general class of polynomial-type sequencesthat these sequences cannot have any S-NBRS, whereas for the sequence ({2n α})n≥1 every
interval is an S-NBRS.

∗The author is supported by the Austrian Science Fund (FWF), Project F5507-N26, which is a part of
the Special Research Program Quasi-Monte Carlo Methods: Theory and Applications.
†
The author is supported by the Austrian Science Fund (FWF), Project F5507-N26, which is a part of theSpecial Research Program Quasi-Monte Carlo Methods: Theory and Applications and Project I1751-N26.

1

26

Construction of normal numbers
Imre Kátai
Eötvös Loránd University, Budapest, Hungary
Given a fixed integer q ≥ 2, an irrational number ξ is said to be a q-normal
number if any preassigned sequence of k digits occurs in the q-ary expansionof ξ with the expected frequency, that is 1/q k . In this talk, we expose new
methods that allow for the construction of large families of normal numbers.
This is joint work with Professor Jean-Marie De Koninck.

27

Automatic and q-multiplicative sequences
through the lens of higher order Fourier analysis
Jakub Konieczny
Hebrew University of Jerusalem & Jagiellonian University

Automatic sequences are among the simplest models of computation. Intuitively speaking, an automatic sequence is one whose n-th term can be computedby a finite device given the digits on n as input. Perhaps the simplest example of an automatic sequence carries the name of Thue–Morse, and is given byt(n) = (−1)s2 (n) , where s2 (n) denotes the sum of digits of n base 2. It is also a
2-multiplicative sequence, meaning that t(n) can be computed as a product ofcontributions which depend on consecutive binary digits of n.
Various uniformity properties of specific automatic sequences have long beenstudied. It was already in 1968 that Gelfond obtained quantitative estimates of
the Fourier coefficients of the Thue–Morse sequence. As a consequence (observed
later by Mauduit and Sarközy), approximately half of the terms in any sufficiently long arithmetic progression have an even sum of binary digits. Behaviour
along more complicated sequences — such as polynomials, Piatetski–Shapiro
sequences or the primes — has been extensively studied by many authors, including Drmota, Mauduit, Müllner, Rivat, Spiegelhofer and others.
With the advent of higher order Fourier analysis, new notions of uniformity
have come to light. Speficically, a sequence f can be construed as uniform (oforder s ≥ 2) — or pseudorandom — if the Gowers uniformity norms kf kU s [N ]become small as N → ∞. The usefulness of this notion stems in large part fromthe fact that the Gowers norms control the count of arithmetic progressions, aswell as other linear patterns. We show that the Gowers norms of the Thue–Morsesequence are very small: ktkU s [N ] = O(N −cs ), where cs > 0 is a constant. An
analogous argument shows the same conclusion for another famous automaticsequence by the name of Rudin–Shapiro, and several other examples. Similarstatements are also true for q-multiplicative sequences (q ≥ 2): in joint workwith A. Fan we show such sequences are either Gowers uniform of all orders,or correlate with a periodic sequence (hence are not Gowers uniform of order2). The talk will be focused on these results, and — time permitting — somerelated results on ergodic theorems weighted by automatic sequences obtained
jointly with T. Eisner.

1

28

ON SUBGRAPHS OF RANDOM CAYLEY SUM GRAPHS
S. V. KONYAGIN, I. D. SHKREDOV

Abstract.
We prove that asymptotically almost surely, the random Cayley sum graph overa finite abelian group G has edge density close to the expected one on every induced
subgraph of size at least logc |G |, for any fixed c > 1 and |G | large enough. For
details see [1].

References[1] S.V. Konyagin, I.D. Shkredov, On subgraphs of random Cayley
sum graphs, European J. Math., 70 (2018), 61–74, arXiv:1710.07320.
S.V. Konyagin
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
konyagin@mi.ras.ru
I.D. Shkredov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
ilya.shkredov@gmail.com

1

29

Uncertainty in Finite Affine Planes
Vsevolod F. Lev

The University of Haifa at Oranim
seva@math.haifa.ac.il
The general uncertainty principle says that a function and its Fouriertransform cannot both be highly concentrated. I will present a recent joint
paper of András Biró and myself, where we establish a number of specificrealizations of this principle for the additive group of the finite affine planeF2p , with p prime.
As an application, I will give an estimate for the largest possible numberof directions in F2p such that a (weighted) set P ⊂ F2p is perfectly uniformly
distributed among the p lines in the direction in question.

30

Multifractal behaviour of the Brjuno function
Bruno Martin (Université du Littoral Côte d’Opale)
CIRM, October 2018
The Brjuno function is 1-periodic, nowhere locally bounded function, introduced by
Yoccoz, because it encapsulates a key information concerning analytic small divisor problem in dimension 1. Its definition is directly related to the continued fraction expansionof a real number. With Stéphane Jaffard we show that this function is multifractal, whichmeans that its pointwise Hölder regularity (in the L1 sense) may widely change from point
to point.

1

31

A DENSITY OF RAMIFIED PRIMESCHRISTINE MCMEEKIN

Abstract. For “nice” number fields K, a certain spin [FIMR13] dependence
relation occurs with density given [McMng] by the formula2n−1 + (mK n + 1)(n − 1)2n n
where n := [K : Q] and mK is a computable and bounded invariant of thenumber field K.
A consequence of this result is a conditional theorem giving a formula forthe density of rational primes exhibiting a prescribed ramified factorization ina number field depending on the prime in question. This density is strictly
between 0 and 1.

References[FIMR13] J. B. Friedlander, H. Iwaniec, B. Mazur, and K. Rubin. The spin of prime ideals. Invent.
Math., 193(3):697–749, 2013.
[McMng] Christine McMeekin. On the asymptotics of prime spin relations, (publication pending).
This paper will become available at https://sites.google.com/a/cornell.edu/christinemcmeekin/research/publications.

Date: July 2018.
1

32

Distribution of short subsequences of theinversive generatorLászló Mérai
Abstract
For an integer t ≥ 3, consider the map ψ : Z∗2t → Z∗2t of the form
ψ(u) =

au + bcu + dfor u ∈ Z∗2t .

with a ≡ d ≡ c + 1 ≡ b + 1 mod 2.
Starting from an initial value u0 ∈ Z∗2t we study the distributionof short subsequences of (un ) defined by the recurrence relation
un+1 = ψ(un )for n = 0, 1, . . .

The main tool is to give bounds on short exponential sums of form
Sh (N ) =

N
−1
X
n=0



2πiun
exp h t,2

gcd(h, 2) = 1.

33

The Rudin-Shapiro sequence and similar
sequences are normal along squares
Clemens Müllner
CNRS Université Claude Bernard Lyon 1
mullner@math.univ-lyon1.fr
September 18, 2018
Recently, there have been discovered some interesting results for subsequences of automatic sequences and especially the Thue-Morse sequence. In
2013 Drmota, Mauduit and Rivat observed that the subsequence along thesquares (t(n2 ))n≥0 of the Thue-Morse sequence (t(n))n≥0 (that can be definedby t(n) = s2 (n) mod 2, where s2 (n) denotes the binary sum-of-digits function)
is a normal sequence on the alphabet {0, 1}, i.e. every block of length ℓ appears
with asymptotic density 2−ℓ .
Another well studied subsequence is ⌊nc ⌋. Recently Spiegelhofer showed that
the Thue-Morse sequence is uniformly distributed along ⌊nc ⌋, where 1 < c <2) and in a joined work by Spiegelhofer and the author that the Thue-Morsesequence is normal for 1 < c < 3/2.
In this talk, we discuss the first generalization of one of these results to moregeneral sequences. One of the most well-known automatic sequences besides
the Thue-Morse sequence is the Rudin-Shapiro sequence (r(n))n≥0 that can be
defined by r(n) = f11 (n) mod 2, where f11 (n) counts the number of occurrencesof the pattern 11 in the binary expansion of n. The Rudin-Shaprio sequencebelongs to the class of block-additive functions modm. Here, a block-additive
function denotes a linear combination of pattern-counting functions fw . Weshow that all block-additive functions modm (with obvious restrictions) arenormal along the squares, which gives a whole class of, easy to compute, normal
numbers.

34

Variations on a theme of K. Mahler
Abstract

Attila PethőDepartment of Computer Science, University of Debrecen,
H-4010 Debrecen, P.O. Box 12, HUNGARY and
University of Ostrava, Faculty of Science,
Dvořákova 7, 70103 Ostrava, CZECH REPUBLIC

For an integer n denote (n)g the sequence of digits of the g-ary representation of n. Mahler [1] proved that the number 0.(1)10 (g)10 (g 2 )10 . . . isirrational for any g ≥ 2. It has many generalizations and refinements. Here
we prove further generalizations. In the first direction we replace the sequenceof powers by weighted sums of elements of a finitely generated multiplicative
semigroup of a number field. In the second direction, the base g is replacedby an algebraic integer. As a byproduct, we prove a Mahler-type result replacing the sequence of powers by a fixed coordinate of solutions of a norm
form equation.

References[1] K. Mahler, On some irrational decimal fractions, J. Number Theory,
13 (1981), 268-269.

35

Tractability properties of the weighted star
discrepancy
Friedrich Pillichshammer
Johannes Kepler University Linz (Austria)
friedrich.pillichshammer@jku.at
Tractability properties of various notions of discrepancy have been intensively studied in the last two decades. In this presentation we consider
the so-called weighted star discrepancy which was introduced by Sloan andWoźniakowski in 1998. The subject of tractability is concerned with thedependence of the weighted star discrepancy on the dimension. Roughly
speaking, the weighted star discrepancy is said to be tractable if the inverseof weighted star discrepancy does not explode exponentially with the dimension.
We show that the Halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights (γj )j≥1 under the
mildest condition on the weight sequence known so far for explicitly constructive sequences. However, there is also a hidden problem in this result
which shall be discussed .
This is joint work with Aicke Hinrichs (JKU Linz) and Shu Tezuka (Kyushu
University).

1

36

An extension of the digital method
based on b-adic integers
Authors:
Speaker:
Affiliation:
Email:

Roswitha Hofer and Ísabel Pirsic
Ísabel Pirsic
Johannes Kepler University Linz
roswitha.hofer@jku.at, isabel.pirsic@jku.at

Keywords quasi-Monte Carlo methods, sequence construction,
digital method, digit expansion, q-adic integers

We introduce a hybridization of digital sequences with uniformly distributed
sequences in the domain of b-adic integers, Zb , b ∈ N \ {1}, by using such sequences as input for generating matrices. The generating matrices are thennaturally required to have finite row-lengths. We exhibit some relations of the‘classical’ digital method to our extended version, and also give several examplesof new constructions with their respective quality assessments in terms of t, T
and discrepancy.

37

Uniform distribution for zeros of random polynomials
byIgor E. Pritsker
Oklahoma State University
igor@math.okstate.edu

Zeros of Kac polynomials (defined as linear combinations of monomials with i.i.d. random coefficients) are asymptotically uniformly distributednear the unit circumference under mild assumptions on the coefficients. We
present several generalizations of such results for lacunary random polynomials, and for polynomials with random coefficients spanned by various bases,
e.g., by orthogonal polynomials. In particular, we show almost sure convergence of the zero counting measures, and quantify this convergence via the
expected discrepancy between the zero counting measures and the limiting
measure.

38

Olivier Ramaré, CNRS / Aix-Marseille Université
Discrepancy estimates for generalized polynomials
Generalized polynomials have been the subject of several investigations in recent times. This talk will present my recent work with Rosewitha Hofer on one
side and with Anirban Mukhopadhyay & G. Kasi Viswanadham on the other side
where the discrepancy of the sequence ([p(n)α]β) is bounded above, where p(x) isa polynomial with real coefficients and α and β are irrational numbers satisfying
certain conditions.

1

39

On the digits of primes and squares
Joel Rivat

Aix-Marseille Université, FRANCE
joel.rivat@univ-amu.fr

I will give a survey of our results on the digits of primes and squares (joint
works with Michael Drmota and Christian Mauduit).

40

Quasi-random graphs and pseudo-randombinary sequences
András Sárközy
ELTE Eötvös Loránd University,
Institute of Mathematics,
Department of Algebra and Number Theory
gykati@cs.elte.hu

The notion of quasi-random graphs was introduced in 1987 by F. R. K.
Chung, R. L. Graham and R. M. Wilson, resp. A. Thomason. It has beenshown that there is a strong connection between this notion and the pseudorandomness of (finite) binary sequences. This connection can be utilized
for constructing large families of quasi-random graphs by considering graphs
defined by a circular adjacency matrix whose first column is a binary sequencewith strong pseudo-random properties. Starting out from this construction
principle one may extend, generalize and sharpen some definitions and resultson quasi-randomness of graphs.

1

41

The Farey graph, continued fractions andSL2-tilings
Ian Short

The Open University, UK
ian.short@open.ac.uk

In the 1970’s, Coxeter studied certain arrays of integers that form friezes
in the plane. He and Conway discovered an elegant way of classifying thesefriezes using triangulated polygons. Recently, there has been a good deal ofinterest in expanding Conway and Coxeter’s ideas and relating them to othermathematical structures, such as SL2 -tilings. Our objective in this talk is todemonstrate how several significant theorems in this field can be explained inan elegant fashion by representing continued fractions geometrically as paths
on an infinite graph known as the Farey graph. We will also put forward a
programme for classifying integer friezes modulo n by working with continued
fractions on certain quotients of the Farey graph.

42

On exponential sums and equations overmultiplicative subgroups in finite field.
Iurii Shteinikov
Let Zp be finite field with p elements and let G ⊂ Z∗p be multiplicative subgroup.
Consider the following exponential sums
S(a, G) =X 2πi age pg∈GWe are interested in non-trivial estimates of the form |S(a, G)| = o(|G|)|. Upper
estimates for |S(a, G)| can be obtained from upper estimates for Tk (G), see [1] where
Tk (G) = |{(x1 , . . . , x2 ) ∈ G2k : x1 + . . . + xk = xk+1 + . . . + x2k }|Several non-trivial estimates for Tk were obtained in [2] using method due to S.A.
Stepanov [3]. In my talk I am planning to present the recent estimate for T3 (G) ([4])
and some other applications.

References[1] S. V. Konyagin, I. E. Shparlinski Character sums with exponential functions andtheir applications Cambridge Tracts in Mathematics, 136, Cambridge University
Press, Cambridge, 1999 ,[2] D. R. HeathBrown, S. V. Konyagin New bounds for Gauss sums derived fromk-th powers, and for Heilbronns exponential sum, Quart. J. Math. 51 (2000),
221235.
[3] S.A. Stepanov The number of points on a hyperelliptic curve over a prime fieldIzv. Akad. Nauk SSSR Ser. Mater. 33 (1969), 11711181.
[4] Brendan Murphy, Misha Rudnev, Ilya D. Shkredov, Yurii N. Shteinikov On the
few products, many sums problem, https://arxiv.org/abs/1712.00410.
[5] Yu. Shteinikov Estimates of trigonometric sums over subgroups and some of theirapplications. Mathematical Notes 98: 3-4 (2015), 667684.
1

43

Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina 8, Moscow,
Russiayuriisht@yandex.ru2

44

THE LEVEL OF DISTRIBUTION OF THE THUE–MORSE SEQUENCELUKAS SPIEGELHOFER
Abstract. The level of distribution of a complex valued sequence b measures how well b
behaves on arithmetic progressions nd + a. Determining whether a given number θ is a levelof distribution for b involves summing a certain error over d ≤ D, where D depends on θ; thiserror is given by comparing a finite sum of b along nd + a and the expected value of the sum.
We prove that the Thue–Morse sequence has level of distribution 1, which is essentially best
possible. More precisely, this sequence gives one of the first nontrivial examples of a sequencesatisfying an analogue of the Elliott–Halberstam conjecture in prime number theory. In
particular, this result improves on the level of distribution 2/3 obtained by Müllner and theauthor.
Moreover, we show that the subsequence of the Thue–Morse sequence indexed by ⌊nc ⌋,where 1 < c < 2, is simply normal. That is, each of the two symbols appears with asymptotic
frequency 1/2 in this subsequence. This result improves on the range 1 < c < 3/2 obtainedby Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved(in particular) that the Thue–Morse sequence along the squares is simply normal. In the
proofs, we reduce both problems to an estimate of a certain Gowers uniformity norm of theThue–Morse sequence similar to that given by Konieczny (2017).

Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner
Hauptstrasse 8–10, 1040 Vienna, Austria1

45

The sum of digits in two different bases
Thomas Stoll

Université de Lorraine
thomas.stoll@univ-lorraine.fr
Let sa (n) denote the sum of digits of an integer n in the base a expansion.
We show that, provided a and b are multiplicatively independent integers,
any positive real number is a limit point of the sequence {sb (n)/sa (n)}n .
We also provide upper and lower bounds for the counting functions of thecorresponding subsequences. This is joint work with R. de la Bretèche and
G. Tenenbaum.

46

Digital questions in finite fields
Cathy Swaenepoel
Université d’Aix-Marseille

The connection between the arithmetic properties of an integer andthe properties of its digits in a given basis produces a lot of interestingquestions and many papers have been devoted to this topic. In thecontext of finite fields, the algebraic structure permits to formulate
and study new problems of interest which might be out of reach in N.
This study was initiated by C. Dartyge and A. Sárközy.
We will devote our interest to several new questions in this spirit:(1) estimate precisely the number of elements of some special sequences of Fq whose sum of digits is fixed;(2) given subsets C and D of Fq , find conditions on |C| and |D| toensure that there exists (c, d) ∈ C × D such that the sum ofdigits of cd belongs to a predefined subset of Fp ;(3) estimate the number of elements of an interesting sequence of
Fq with preassigned digits.
We notice that the notion of digits in Fq is directly related to the notionof trace which is crucial in the study of finite fields.

47

Discrepancy bounds for β-adic Halton sequencesJörg Thuswaldner
University of Leobenjoerg.thuswaldner@unileoben.ac.at
Van der Corput and Halton sequences are well-known low-discrepancy sequences.
In the 1990ies Ninomiya defined analogs of van der Corput sequences for β-numeration and proved that they also form low-discrepancy sequences provided that β isa Pisot number. Hofer, Iacó, and Tichy define β-adic Halton sequences and showthat they are equidistributed for certain parameters β = (β1 , . . . , βs ).
In this talk we give discrepancy estimates for β-adic Halton sequences for which
the components βi are m-bonacci numbers. Our methods include dynamical andgeometric properties of Rauzy fractals that allow to relate β-adic Halton sequences
to rotations on high dimensional tori. The discrepancies of these rotations can thenbe estimated by classical methods relying on W. M. Schmidt’s Subspace Theorem.

48

Title of the talk: NORMALITY and RANDOMNESS
Name: Robert Tichy
Address:
Technische Universität Graz
Institut für Analysis und Zahlentheorie
Steyrergasse 30
8010 Graze-mail: tichy@tugraz.at
We focus on various criteria introduced by Donald Knuth to describe (pseudo)
random properties of sequences. As a starting point we consider the classicalconcept of normal numbers in q-ary digital expansion. We present probabilistic
results as well as various constructions for normal numbers. Furthermore, weestablish recent results on efficient constructions for absolutely normal numbers(i.e. numbers which are normal in any base q). We analyze the speed of convergence to normality as well as the computational complexity of the construction
algorithms. In a final section, we obtain normality results with respect to moregeneral numeration systems.

1

49

An elementary approach to Somos-4 sequences
Alexey Ustinov

Pacific National University, Institute of Applied Mathematics(Khabarovsk Division, Far-Eastern Branch of the Russian
Academy of Sciences) RUSSIA
ustinov.alexey@gmail.com

A sequence Somos-4 is defined by initial data s0 , s1 , s2 , s3 and fourthorder recurrence
sn+2 sn−2 = αsn+1 sn−1 + βs2n .
Usually properties of this sequence are studied by means of elliptic functions. The talk will be devoted to the new elementary approach to Somos-4
sequences. Hopefully it will be suitable for higher-rank Somos sequences
corresponding to curves of higher genus.

50

Talk at UDT-2018, CIRM, Luminy, France, 1-5 October 2018.
GROWTH AND GEOMETRY IN SL2 (Z) DYNAMICSALEXANDER P. VESELOV (LOUGHBOROUGH, UK)

Usual discrete dynamics can be considered as the action of the group of
integers. What happens if we replace Z by SL2 (Z)?
There is a classical example of such dynamics goes back to remarkable
work by Andrei A. Markov (1880), who described the solutions of the Diophantine equation
x2 + y 2 + z 2 = 3xyz
(known now as Markov triples) as an orbit of SL2 (Z). These triples surprisingly appeared in many areas of mathematics: initially in arithmetic,
but more recently in hyperbolic and algebraic geometry, the theory of Teichmüller spaces, Frobenius manifolds and Painlevé equations.
Another example of such dynamics appears in the description of the valuesof a binary quadratic form Q(x, y) = ax2 +bxy+cy 2 with integer coefficients,
the problem going back to Gauss. About 20 years ago John H. Conway proposed a ”topographic” approach to this problem, using the planar trivalenttree, which can be considered as a discrete version of the hyperbolic plane.
The same approach can be applied to general SL2 (Z) dynamics, and inparticular to Markov dynamics as well. The growth of the corresponding
numbers depends on the paths on such tree, which can be labelled by the
points of real projective line.
I will discuss some results about the corresponding Lyapunov exponents
found jointly with K. Spalding and A. Sorrentino, using the known links
with the hyperbolic geometry.

51

On the maximum order complexity of subsequences of
the Thue-Morse and Rudin-Shapiro sequence alongsquaresArne Winterhof (Austrian Academy of Sciences, joint work with Z. Sun)
Automatic sequences such as the Thue-Morse sequence and the Rudin-Shapiro
sequence are highly predictable and thus not suitable in cryptography. In particular, they have small expansion complexity. However, they still have a large
maximum order complexity.
Certain subsequences of automatic sequences are not automatic anymore
and may be attractive candidates for applications in cryptography. We showthat subsequences along the squares of certain pattern sequences including theThue-Morse sequence and the Rudin-Shapiro sequence have also large maximum
order complexity but do not suffer a small expansion complexity anymore.

1

52

METRIC DISCREPANCY WITH RESPECT TO FRACTAL MEASURES
Agamemnon Zafeiropoulos
TU Graz, Austria
zafeiropoulos@math.tugraz.at
Let (nk )∞k=1 be a lacunary sequence of integers. In joint work with N. Technau, we showthat if µ is a probability measure on [0, 1) such that |bµ(t)| ≤ c|t|−η , then for µ-almost allx, the discrepancy DN (nk x) satisfies
DN (nk x) ≪ N −1/2 (log log N )1/2 ,N → ∞.