Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

DISCREPANCY estimatesfor generalized POLYNOMIALS
Olivier Ramaré
September 28, 2018

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Please Publish in ACADEMIC Journals!
likeNWEJM

North Western European
Journal of Mathematics

Walter Craig, Sinnou David, Jean-Pierre Demailly, FlorianLuca, Kaisa Matomäki, OR, Jie Wu,. . .

FUNCTIONES ET
APPROXIMATIO

Little brotherof Acta ArithmeticaEuclid Project

Jerzy Kaczorowski, Julian Musielak, Leszek Skrzypczak,. . .
Several years of existence, Referenced in Mathscinet and Zentralblatt
HRI Lecture Notes Series, IMSc (Chennai) Lecture Notes in Mathematics

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Initiation
In 2008 (Green & Tao, 2012):
The Möbius function isls
d Polynomia
strongly orthogonal to Generalize nilsequences

Generalized Polynomials = Bracket Polynomials
Polynomials like tn 3?

2 ` n7 rn?

5 ` n2 s `?
πu

§ rx s = integer part (floor) of x.
§ tx u = fractional part of x.
Strange beasts: rpn ` 1qπs ´ rnπs ´ 3, 23 tπnu ` 13 r2tπnus

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

New information about µpnq, but what is it?
Project: Using Generalized Polynomials§ Chennai 2011: Working group with Anirban
Mukhopadhyay
We want precise and flexible informations
With tαnu, we have Fourier analysis and localisation. We seek similar geometrical understanding.

§ Linz 2015: Roswitha Hofer interested in subsequences.

§ 2016: G. Kasi Viswanadham joined in.

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Equidistribution
Theorem (Folklore)n ÞÑ tβrαnsu is equidistributed modulo 1
iff 1, α, αβ are linearly independent / Q.

Theorem ((Veech, 1971), (Håland, 1993))
P px q “ x d ` a1 x d ´1 ` ... ` ad , αβ ‰ 0§ When a1 , ..., ad rational, tβrαP pnqsu
equidistributed iff 1, α, αβ lin. indep./Q.
§ Else tβrαP pnqsu equidistributed.

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Diophantine Backgroundpt ` 1q is an irrationality measure of γ P R whenmin |m|t `ε }mγ} "ε,γ 1.

mPZzt0u

In which case, we have (via Erdős-Turáninequality):
DN pnαq !α,ε N ´1{t `ε .
Corresponding notion for a pair of real numbers?

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

We follow (Niederreiter, 1973):pγ, δq P R2 is of finite type t when`˘t `εminp1 ` |m|qp1 ` |n|q}mγ ` nδ} "ε 1.
pm,nqPZ2 ztp0,0qu

1. Let γ, δ real algebraic numbers, 1, γ, δ are lin.
indep./Q. (Schmidt, 1967): pγ, δq is of finite
type 1.
2. For almost all pγ, δq P R2 , the pair pγ, δq is offinite type 1.

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

To accomodate usage, Niederreiter proposed to us:pγ, δq is of weak cotype t 1 ě 1 when, for any ε ą 0,min

1p|m| ` |n|qt `ε }mγ ` nδ} "ε 1.

pm,nqPZ2 ztp0,0qu

weak cotype t 1 ùñ finite type t 1 |m| ` |n| ď p1 ` |m|qp1 ` |n|q(Hata, 1992):min
2pm,nqPZ ztp0,0qup|m| ` |n|q7.0161 }mπ ` n log 2} " 1.
Other examples??

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Discrepancy, ITheorem (R. Hofer & OR (2015))
Let α, β real numbers§ 1, α, αβ lin. indep./Q (automatic).
§ pα, αβq and pβ, 1{αq of finite type t.
Then, for every ε ą 0,DN prnαsβq !α,β,ε N ´1{p3t ´2q`ε .
That’s 1{N 1´ε when t “ 1!

Optimal??

With weak cotype, a hypothesis either on pα, αβq or on pβ, 1{αq is enough.

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

CorollaryFor almost all pairs of real numbers pα, βq inthe sense of Lebesgue measure we have forevery ε ą 0 that DN prnαsβq !α,β,ε N ´1`ε .

CorollaryDiscrepancy of prn{πs log 2q and prn{ log 2sπq is! N ´0.052498 .
Omega Estimates??

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Equidistribution, II(Bergelson & Leibman, 2007)§ Dynamical understanding of GeneralizedPolynomials.
§ Description of which measure makes aGeneralized Polynomials well-distributed.
§ The strange beasts may be more telling . . .
Ohio State University!

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Discrepancy, II
Back to India: let us look at tβrαn2 su which I thought would be very difficult.

Theorem (A. Mukhopadhyay & OR & G. Kasi
Viswanadham – 2018)
Let pα, αβq of finite type t ě 1. For  ą 0,DN prP pnqαsβq !,α,β,d N´ d ´12´22´d `2p2t `1q`7t `2`.

P pnq monic degree d ě 2.
11

7d “ 2 Ñ! N ´ 11t `4 , d “ 3 Ñ! N ´ 10t `4 , d “ 4 Ñ! N ´ 92t `40 .
We do not have metric results, though that may change. Or Omega estimate

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Lemma ((Vinogradov, 1927))
Let α of irrationality meas. t ` 1. For any  ą 0.
DN pP pnqαq !,d ,t N

´2dp1`t q2d`1` N ´ 2d´1 `P pnq monic degree d ě 2.
Ancestors? d “ 2 in (Behnke, 1922)
Successors???
With (Vinogradov, 2004), the exponent becomes ´E whereE"1.
pd 2 log d qpt ` d 2 log d q

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Elements of proofFor τ P R, let fτ px q “ e pτtx uq.
For δ ą 0,1gτ,δ px q “1r´δ,δs ˚ ¨ ¨ ¨ ˚ 1r´δ,δs ˚ fτ px q,
p2δqr

Lemma (gτ,δ approximates fτ )For any tun uně1 , and any N we haveÿn ďN|fτ pun q´ gτ,δ pun q| ! Nr δ` Nr 2 δ|τ|` NDN pun q.

First key

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof Referencesˆĝτ,δ pk q “Lemmasin 2πk δ
2πk δ˙rLet τ, δ ą 0, p ą 1, thene pτ ` k q ´ 1
2πi pτ ` k qÿ|ĝτ,δ pk q|p !p 1.

k PZ

Uniform in |τ| ď 1{p2δq+ more usual lemmas on exponential sums withpolynomial phase.

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Skeleton§ βrαP pnqs “ βαP pnq ´ βtαP pnqu.
§ Erdős-Turán inequality.
§ NDN pr
ˇ P pnqαsβq !
ˇHNÿ
´1
ˇ
ÿ 1 ˇˇ ÿˇαphβ´kqe pP pn qqˇ `εˇ ĝ´h β,δph q pk q
ˇh ˇk PZ
h “1n“0Explicit dep. in h and k !

§ We use only process A. But we haveexponential sums with parameters!

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Behnke, H. 1922.
Über die Verteilung von Irrationalitäten mod. 1.
Abh. Math. Sem. Univ. Hamburg, 1(1), 251–266.
Bergelson, Vitaly, & Leibman, Alexander. 2007.
Distribution of values of bounded generalized polynomials.
Acta Math., 198(2), 155–230.
Green, B., & Tao, T. 2012.
The Möbius function is strongly orthogonal to nilsequences.
Ann. of Math. (2), 175(2), 541–566.
Håland, Inger Johanne. 1993.
Uniform distribution of generalized polynomials.
J. Number Theory, 45(3), 327–366.
Hata, Masayoshi. 1992.
Improvement in the irrationality measures of π and π2 .
Proc. Japan Acad. Ser. A Math. Sci., 68(9), 283–286.
Hofer, R., & Ramaré, O. 2015.
Discrepancy estimates for some linear generalized monomials.
Acta Arith., 173(2), 183–196.
Koksma, J. F. 1974.
Diophantische Approximationen.
Springer-Verlag, Berlin-New York.
Reprint.
Mukhopadhyay, A., Ramaré, O., & Viswanadham, G. Kasi. 2017.
Discrepancy estimates for generalized polynomials.
Monatshefte für Mathematik, 1–14.
Niederreiter, H. 1973.
Application of Diophantine approximations to numerical integration.
129–199.

Initiation Equidistribution Diophantine Background Discrepancy, I Equidistribution, II Discrepancy, II Elements of proof References

Niederreiter, Harald. 2009.
On the discrepancy of some hybrid sequences.
Acta Arith., 138(4), 373–398.
Schmidt, Wolfgang M. 1967.
On simultaneous approximations of two algebraic numbers by rationals.
Acta Math., 119, 27–50.
Veech, William A. 1971.
Well distributed sequences of integers.
Trans. Amer. Math. Soc., 161, 63–70.
Vinogradov, I. M. 1927.
Analytischer Beweis des Satzes über die Verteilung der Bruchteile eines Polynoms.
Leningrad, Bull. Ac. Sc. (6), 21, 567–578.
Vinogradov, I.M. 2004.
The method of trigonometrical sums in the theory of numbers.
Mineola, NY: Dover Publications Inc.
Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport, Reprint of the 1954translation.