Newsletter of the EMS – Manuscript

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Pierre Liardet (1943–2014) in memoriam
Guy Barat, Peter J. Grabner, & Peter Hellekalek
1

Introduction

Words and numbers cannot describe a man. The three authorsof this paper had a special relationship with Pierre Liardet:as a former PhD student, as former collaborators, and also as
friends. As such, we will try to give the reader an idea of
the numerous contributions of Pierre to number theory andrelated fields and further, of the fine man behind these results.
Pierre Liardet was born in Gardanne, close to Aix-en-Provence, on March 20, 1943. After receiving his baccalauréat
en mathématiques et technologie at the Lycée Saint-Éloi in
Aix-en-Provence in 1961, he studied mathematics at the University in Marseille finishing with his doctorate in 1970. Afterwards, he obtained a position as a “Maître Assistant” at
the Université de Provence and submitted his Thèse d’État in1975. Soon afterwards he became a professor and later “Professeur de première classe” at this university. From 1990–
2010 he directed the team “Dynamique, Stochastique et Algorithmique” (DSA).
He was an editor of the “Journal de Théorie des Nombresde Bordeaux” and of “Uniform Distribution Theory”. In 2005
he organised the biannual international conference on number theory “Journées Arithmétiques”. Pierre liked to traveland visited many countries all over the world. This is reflected by the fact that he had coauthors from many countries: Austria, Canada, China, France, Hungary, Japan, theNetherlands, Poland, Romania, USA. He was professor emeritus since 2012 and was still active in research until his lastdays. He passed away on August 29, 2014.
Even if he turned his interests rapidly to analysis, Pierrethought as an algebraist, as he was more interested in the
structures governing mathematical objects and in the operations on them, than in the objects themselves. In this sense he
was an inheritor of Bourbaki’s tradition.

main interest of this research were rational transformations,which leave certain sets of algebraic numbers stable (cf. [L3]).
One of his results was published in [L1] and concerned a stability property of the set Θn of algebraic integers with exactly
n conjugates of modulus greater than 1, and all the other conjugates of modulus smaller than 1 (the case n = 1 are thePisot-Vijayaraghavan numbers).
Theorem 1. Let f be a non-constant rational function definedover an algebraic extension of Q, and E be the complementin Θn of a bounded set. Then∀θ ∈ E : f (θ) ∈ Θn ⇒ ∃m ∈ N : f (x) = ±xm .
After finishing his doctorate he continued working in algebraic number theory and algebraic geometry until his Thèsed’État [L6] in 1975, devoted to the stability of algebraic properties of sets of polynomials and rational functions (cf. [L9,L10]). Two outstanding results from this work were published, namely the disproof of a conjecture of W. Narkiewicz[L4, L8], and the proof of a conjecture of S. Lang. The latter is formulated in terms of intersections of abelian varieties
with algebraic curves in [L5]; we give the more elementary
formulation from [L7].
Theorem 2. Let Γ0 be a finitely generated subgroup of C∗
and let Γ = {x ∈ C∗ | ∃n ∈ N, n , 0 : xn ∈ Γ0 }. If P(X, Y) ∈C[X, Y] is such that P(α, β) = 0 for infinitely many α, β ∈ Γ,
then there exist non-zero integers u, v and a, b ∈ Γ, such thatP(axu , bxv ) is identically zero.
After this fruitful period, he turned his interest mostly into
ergodic theory, dynamical systems, and uniform distributionof sequences.
3

Ergodic theory

Pierre started with reading the most important authors and assimilated the works of Conze, Furstenberg, Kakutani, Keane,
Klaus Schmidt, and others. He published papers in “pure” ergodic theory between 1978 [L11] and 2000 [L36]. The first
paper where Pierre grappled with ergodic theory is [L11]. It isnoteworthy that this paper already deals with the main themesof his further research articles.
Let X be a compact metrisable space, G a compact metrisable group, and ϕ : X → G a continuous map. For T a trans
formation on X, one can define the skew product XϕG, T ϕ
byFigure 1: Pierre hiking in the French Alps2

Algebraic number theory and algebraic geometry

Pierre Liardet wrote his PhD thesis [L2] under the advice of
Gérard Rauzy on a subject from algebraic number theory. The

T ϕ : X × G → X × G,

T ϕ (x, g) = (T x, g · ϕ(x))

and ask about relations between properties of (X, T ) and prop
erties of XϕG, T ϕ . Skew products both furnish an inexhaustible source of examples, allow to describe by isomorphism several flows associated to sequences (there are numerous such examples in Pierre’s work, see for instance [L42])
and illustrate the general idea that a good way to understandNewsletter of the EMS – Manuscript

mathematical objects is to describe how to act on them. For
those reasons, Pierre had a continuous and deep interest in
this type of product systems. In this first paper, beginningwith considerations from uniform distribution theory, Pierregave characterizations of ergodicity and weak mixing of large
families of skew products, extending results of Furstenberg
and Conze; he could also mold a theorem of Veech in a more
precise form.
The use of the ergodicity of certain types of skew products was one of Pierre’s central techniques to prove results
for certain sequences like irregularities of distribution (seeSection 4). Usually, these sequences were shown to be the orbits of points in some other space under an odometer (adding
machine transformation, see Section 5), which translates tothe study of a group rotation. Further investigations on skew
products can be found in [L13] and [L36]. K. Schmidt (probably) introduced him to Zd -actions, on which he published[L28] and [L30]. His deep interest on spectral approaches
originates from here.
The paper [L31] deals with the speed of convergence inBirkhoff’s ergodic theorem. Let T be a homeomorphism of
the compact metric space X and C0 (X) be the space of all
real-valued continuous functions f having zero integral withrespect to a fixed T -invariant aperiodic measure λ. Then there
exists a Gδ set in C0 (X) such that distributions of the randomP
variables c1n 0≤ j<n f ◦T j are dense in the set of all probability
measures on the real line, where cn ↑ ∞ and and cn /n → 0.
The second part of the paper is devoted to irrational rotations
Rα on the torus T = R/Z. Functions defined on T are often called cocycles in this context. Moreover, a cocycle F isa coboundary if there exists G such that F = G − G ◦ Rα .
P
(Hence, coboundaries F make the sums F ◦ T n telescopic).
The last theorem in [L31] shows that if α has bounded partial
quotients, then there exists a dense Gδ subset of the set of absolutely continuous cocycles with zero integral, which are notcoboundaries. Similar results are shown when the sequenceof partial quotients is not bounded, and convergence rates of(S n )n are given for several subspaces of cocycles in the latter
case.
Studies in the same flavour have been done for Anzai’s
skew product extensions of the two-dimensional torus, that
Pierre investigated in [L21] and [L25]. The monograph [L21]
contains an elaborate study of cocycles which are either absolutely continuous or step functions, and a general discussion on group extensions, which yield families of interesting examples. Pierre supervised four PhD theses on subjects related to his interests in ergodic theory: J. Mouline(1990), Y. Lacroix (1992), E. Olivier (1997), and C. GuilleBiel (1997).
4

Uniform distribution

Pierre’s contributions were mainly focused upon two topics,
sequences generated by certain digit expansions of real numbers, and by the application of notions of ergodic theory. Two
keywords in this context are digital sequences and irregularities of distribution.
The dynamical systems approach that Pierre had chosen
to study statistical properties of a sequence ω = (xn )n in a

Page 2

Figure 2: Pierre with Oto Strauch at a conference in Smolencein 2012compact metrisable space X, with X = [0, 1) s , s ≥ 1, the sdimensional torus as the most important case, proceeds in two
steps: first, one has to find the dynamical system behind thegiven sequence ω, and, in the second step, one has to employ
properties of this system to derive distribution properties ofthe sequence.
To illustrate this approach, let ω = (xn )n≥0 be a sequenceon the torus [0, 1) s . For a (Lebesgue-) measurable subset A of[0, 1) s , let λ s (A) denote its Lebesgue measure. The quantity
RN (A, ω) = ] {n, 0 ≤ n < N : xn ∈ A} − Nλ s (A)is called the local discrepancy of the first N points of ω forthe set A or, for short, the remainder of A.
For a given sequence ω, is there any hope that the remainders RN (A, ω) stay bounded in N? In an impressive seriesof papers, Wolfgang Schmidt investigated this question ofbounded remainder sets by techniques from metric number
theory. Here, two questions are central:(i) for sets A with bounded remainder, determine the setof possible volumes λ s (A) (the so-called admissible volumes), and(ii) identify all bounded remainder sets, at least in the casewhen they are s-dimensional intervals.
Question (i) may be related to topological dynamics and ergodic theory (cf. [3, 8]). From this approach one obtains theresult that the set of admissible volumes stems from eigenvalues of an isometric operator. The main contributions of Pierrein this context concern general versions of the coboundary
theorems that are involved here. This allowed Pierre to giveat least partial answers to the bounded remainder sets problem in case of certain sequences (see [L16]). This area is still
active, which is shown by the recent breakthrough in [4].
Question (ii) is much more difficult to attack and results are
known only in very special cases for ω and for A. For example, this is the case for Kronecker sequences (nα mod 1)n≥0
in [0, 1) s (see [5] for s = 1 and [L16] for s > 1). An important result of Pierre deals with polynomial sequences (p(n)
mod 1)n≥0 (cf. [L16]). In the same paper, as an example tohis general approach, he has shown the following. Let α beirrational and let sg (n) be the sum of digits of n to the baseg ≥ 2. Then the only intervals I of the 1-torus [0, 1) which arebounded remainder sets for the sequence (αsg (n))n≥0 in [0, 1)
are the trivial ones, that is to say, |I| = 0 or 1.
In the early years of Pierre’s examination of uniform distribution and discrepancy theory, he gave an ingenious andNewsletter of the EMS – Manuscripttransparent new proof of W. Schmidt’s lower bound for thediscrepancy (i.e., the supremum DN (ω) of the local discrepancies for all intervals A ⊂ [0, 1) ) of a sequence ω = (xn )n≥0
in [0, 1),
DN (ω) = supAlog N
RN (A, ω)≥C, for infinitely many N,
NN

31
refining Schmidt’s constant C = 66 log
4 to 40 log 5 . Unfortunately, this result was only published in an institute report[L12]; the proof can be found in [2, pp. 41–44]. Pierre mentioned at several occasions that it was his dream to find thebest possible constant C, and that its value should be an eigenvalue of an operator related to the problem.
A second series of Pierre’s contributions to the theory of
uniform distribution of sequences concerned the interplay between sequences related to numeration systems and the qualitative behavior of such sequences. Historically, these investigations started in close cooperation with Coquet [L15] on notions of (statistical) independence of sequences and then proceeded to measure-theoretic aspects of numeration systems
in cooperation with Grabner and Tichy [L34], and with Baláž
and Strauch [L54].

5

Systems of numeration and digital expansions

A significant part of Pierre’s achievement concerns numeration, sequences related to diverse representations of real, complex or natural numbers, and the underlying dynamical systems. Those themes traversed Pierre’s mathematical life for25 years, and he was able to fruitfully exploit their interplay.
One can emphasise three seminal papers in this area.
The first one - [L18] - was very important for him, scientifically as well as humanly. During the seventies and eighties, there was an intense interest on statistic properties of sequences, originated in early works of Wiener. Let W be theset of complex sequences u = (un )n such that the correlations
1 X
um+n unN→∞ Nn<Nγu (m) = lim

exist for all m ∈ N. Extended to negative integers by γu (−m) =γu (m), the correlations (γu (m))m form a positive definite sequence, hence are the Fourier coefficients of some Borel measure on the one-dimensional torus T by the Bochner-Herglotz
theorem. This measure is called the spectral measure of u. Investigations on this measure yielded to several concepts andresults on pseudo-random sequences. Bésineau, Coquet, Kamae, Mendès-France, M. Queffélec, Rauzy, Rhin and others
made several contributions to the subject. Pierre was both interested and engaged in that framework and published [L14],
where the authors used their abilities in ergodic theory toshow several results on spectral disjointness. Pierre publisheda continuation [L15] of this work, which contains among otherthings in a more general setting the following result:
Theorem 3. Let θ be a q-normal real number ((qn θ (mod 1))nbeing uniformly distributed) and P a real polynomial. Thenthe two sequences (qn θ (mod 1))n and (P(n) (mod 1))n are
spectrally disjoint.

Page 3

Meanwhile, Coquet passed away and it affected Pierre alot. Coquet’s wife communicated Pierre the private drafts ofher husband and Pierre wrote a synthesis of them in [L18].
That included the redaction of some proofs, that had only
been sketched by Coquet. In this paper, one encounters forthe first time in Pierre’s work the notion of systems of numeration, that is an increasing sequence of integers (Gn )n with
G0 = 1. Any integer m ∈ N can be expressed uniquely asXXm=
εk (m)Gk , provided that ∀n :εk (m)Gk < Gn .
k≥0

k<n

Pierre developed the theory of systems of numeration in [L29],
which is his most cited paper and the second one we emphasise. Here, the closure of the set of representations is considered, namely the set

KG = (εk )k≥0 ; ∀n : ε0 (m)G0 + · · · εn−1 (m)Gn−1 < Gn .
An extension of the addition τ : m 7→ m + 1 is constructed
on KG , which yields a dynamical system (KG , τ), called theodometer. Surjectivity, continuity and minimality are discussed. Special and important examples are Ostrowski systems, based on continued fraction expansions, where Gn = qn ,for α = [0; a0 , a1 , . . .] ∈ [1/2, 1), and qpnn = [0; a0 , a1 , . . . , an ],and sequences (Gn )n satisfying linear recursions of particular
types. The study of odometers was pursued in several papers,
especially [L35] and [L40]. Pierre together with several coauthors also published an overview on numeration in [L45],
where they introduced a notion of fibred numeration systemafter Schweiger [7].
Arithmetical functions related to systems of numeration havebeen extensively studied by the Austrian school and Pierrecollaborated in several papers on the subject. We cite [L33],[L39] which he liked very much, and [L42, L47, L49, L50].
Some of those papers were devoted both to arithmetical functions and odometers. Pierre liked to construct dynamical systems and to investigate them, deciphering their structures, comparing them through conjugacy and other accurate notions of
isomorphism.
The third paper we want to emphasise is [L20]. Let us
consider a trigonometric polynomial
P(eiθ ) =NXn einθ

(1)

n=1

√
with n ∈ {−1, 1}. Parseval Theorem shows that kPk∞ ≥ N
and it follows from results of Salem and Zygmund that the
expected order of magnitude for kPk∞ is (N log N)1/2 for almost all such sequences n ∈ {−1, 1}. In 1951, Shapiro, in
his thesis, gave an inductive construction of a sequence (n )nsuch that |P(eiθ )|  N 1/2 . Rudin rediscovered the sequence in1959 and it has been since called Rudin-Shapiro sequence. Itturns out that n has a simple digital interpretation: it counts
the parity of the number of appearances of the string “11” inthe binary expansion of n.
In [L20], Allouche and Liardet extended this result in a
way which was typical of Pierre’s thinking. First counting
occurrences of subwords of the type a0 a1 · · · ad with a0 ad ,00 (generalised Rudin-Shapiro sequences), they focus on theproperties of such sequences and introduce a general abstract

Newsletter of the EMS – Manuscript

Page 4machinery including those particular cases. A notion of chained
map on an alphabet A taking its values in a compact metrisable group G is introduced and investigated, especially throughan appropriate matrix formalism. Those maps f satisfy the relationf (αβγ) = f (αβ) f (β)−1 f (βγ)

Pierre’s part of this survey was a very general but still accessible and pedagogical description of the cutting-stacking
construction for interval exchange maps which originated in
Rokhlin’s work and was developed further by Kakutani.
Over the years, he had intensive cooperations on cryptographic and algorithmic applications, where he contributed
his mastery of dynamical systems. He always intended tofor non-empty words α, β and γ and are proven to be qmake the descriptions of these notions and techniques underautomatic if A = {0, 1, . . . , q − 1} and if G is finite. Normstandable for his readers, but he was also interested in thebounds for functions of the form (1) are obtained, where theapplications themselves. It is noteworthy that his son Pierreexponential function is replaced by an irreducible represenYvan is working in computer science, and Pierre was regutation of the group G. In a further section, the generalised
larly discussing scientific issues with him, even if they did
Rudin-Shapiro sequences are studied for themselves from a
not formally collaborate. Pierre-Yvan likes to narrate how
dynamic point of view and the authors prove the following:
his father helped him for a crucial counting that he neededfor [6]. Father and son regularly climbed the Sainte-Victoire Theorem 4. Let u : N → U be a generalised multiplicative
Cézanne’s mountain - and it was the occasion of many mathRudin-Shapiro sequence. Then u has a correlation function
ematical discussions. Pierre-Yvan introduced his father to his
and its spectral measure is the Lebesgue measure.
cryptographic questions and Pierre used his theoretic knowThe paper ends with the following announcement: “In aledge to tackle the problem.
forthcoming paper we shall study the flows associated to chained
Obvious applications of dynamical systems are random numsequences, proving that they are (except for degenerate cases)
ber generators, which are used in different branches of comstrictly ergodic and can be obtained as group extensions of aputer science. Evolutionary algorithms are special stochastic
a-adic rotation; we shall also give the spectral study of thesesearch algorithms, which allowed Pierre to introduce his viewflows.”. Eighteen years later, this program was completedof dynamical systems to optimise the implementation. The rein [L52].
sults of these efforts are the papers [L41, L44, L51]. The two
Other number representations, such as continued fractions
papers [L55, L56] propose algorithms for the generation ofand Engel series, also caught Pierre’s attention. Methods fromrandom selections of k elements out of n. These contributions
ergodic theory, especially skew products, are used in [L17] togive a considerable improvement in the rate of convergenceprove uniform distribution of the numerators pn and denomtowards uniformity. Similarly, a randomisation in the coninators qn of the convergents in residue classes modulo m.
struction of the implementation of finite fields by tower fields
In [L22] the set E(α) consisting of all real numbers β suchis proposed in [L61], in order to make the AES cryptosystemthat (βqn (α))n tends to 0 modulo 1 is studied depending on
less vulnerable to side-channel attacks.
diophantine properties of α.
A further branch of Pierre’s research with application to
In [L32] (infinite) automata are constructed, which comcomputer science is the prediction of binary sequences withpute the partial quotients of f (x) from the partial quotients
automata. This started with Annie Broglio’s PhD thesis inof x for certain rational
functionsf.
For
instance,thepar1991, which led to the paper [L23], and continued with [L38].
√3
tial quotients of 2 can be computed by such an automaton.
Pierre supervised three PhD theses on subjects related toThis work was continued in [L37], where (infinite) automata
applications in computer science, cryptography and automata:are given, which compute the Engel series expansion from theA. Broglio (1991), B. Peirani (1994), and S. Rochet (1998).
continued fraction and vice versa.
Pierre’s work in that context was the basis for seven doc7 Concluding remarks
toral theses: D. Barbolosi (1988), C. Faivre (1990), P. Stambul (1994), G. Barat (1995), N. Loraud (1996), M. Doudékova- Pierre Liardet left a lasting influence in all research areas hePuysdebois (1999), and I. Abou (2008). Several of the above
was working in. His results on the intersection of algebraic
mentioned results were generalised and extended in their work. curves with abelian varieties solving a conjecture of Lang
6

Applications

It was very typical for Pierre and his point of view of mathematics, and science in general, that he was trying to make hisknowledge, especially of ergodic theory, accessible to people working in other areas of mathematics, but also to more
applied scientists, especially computer scientists.
At the Conference on Uniform Distribution Theory in Marseille in January 2008, Pierre brought up the idea to makemethods from ergodic theory accessible for, and more popularamongst people working on low-discrepancy sequences. This
led to many informal discussions, to an educational workshop“Dynamical Aspects of Low-Discrepancy-Sequences” held in
Linz in September 2009, and finally, the survey article [L57].

were amongst the first results in this direction, see [1].
After he left the area of algebraic geometry and started towork in ergodic theory, he was influenced by the work of Coquet and Mendès France. He contributed several importantworks on the dynamics of the orbit closure of sequences defined by digital functions. This had been of interest from workof Wiener and Mahler from 1927 and still provides interesting examples of dynamical systems. Besides his studies on
the interplay of number theory and ergodic theory he contributed to the development of ergodic theory itself by deep
investigations about skew products. The propagation of thestudy of odometers and other digital representations as dynamical systems was his personal concern. By the dissemination of this idea he influenced – directly and indirectly –Newsletter of the EMS – Manuscript

Page 5many young mathematicians, especially in France and Aus[L4]tria. The influence of his driving idea of finding dynamical[L5]
systems behind many mathematical questions even led him
to his cooperations with computer scientists. Reading the pa[L6]pers discussed in Section 6 one can immediately pin down
his contribution, namely making apparent the dynamical system behind the seemingly unrelated problem originating fromcomputer science.
[L7]
Pierre Liardet had great influence on the current development of the theory of uniform distribution of sequences modulo one. He was one of the innovators to apply concepts[L8]
and results of ergodic theory to open problems concerning
irregularities of distribution of sequences, which has allowed[L9]
to advance after about two decades of a stand-still after the
ground-breaking results of K. F. Roth and W. Schmidt to thisfield. Further, his insistence on what he called “the dynamic
point of view” of sequences has changed the attitude with
[L10]
which various types of sequences are studied nowadays andhas helped to develop a broader view of the phenomena that
[L11]
appear. Pierre was also one of the leading researchers to bring
the theory of automata, ergodic theory, and the theory of uniform distribution together to study the fine structure of various[L12]
types of sequences.
Pierre liked to meet colleagues and organised several meet[L13]
ings and conferences, among others the Journées Arithmé[L14]tiques in Marseille in 2005, and several meetings at the CIRM
on number theory, ergodic theory, uniform distribution and[L15]
their interplay. He also created more specialised events, likethe “Journées de numération”, which has become a regular[L16]
meeting of researchers working in various aspects of digital
expansions, ranging from automata theory to topology and[L17]diophantine approximation. The last conference in this series
took place in Nancy in June 2015. Indeed, during the last
[L18]
thirty years, Numeration has become a dynamical field andPierre was one of the leading figures of the group.
We have given an overview of Pierre’s scientific work, achievements, and also sketched his influence in several fields. Yet[L19]
besides his constant interest in science, especially mathemat[L20]
ics but also medicine and computer science, he was a loving
husband to Josy, a father to Pierre-Yvan, Frédéric, who sur[L21]
vived his father only by four months, and Christine, and acaring grandfather to his grandchildren. Pierre was a goodfriend and colleague to many of us. He could be missionary[L22]
for his mathematical point of view, but he was always open
[L23]
for discussion and dialogue. His mathematical thoughts andhis approach to doing mathematics have influenced many ofus, especially the three authors. We will miss his constant
input and encouragement.
[L24]
Publications of Pierre Liardet
[L1]

[L2]
[L3]

M. Ventadoux and P. Liardet, Transformations rationnelles laissantstables certains ensembles de nombres algébriques. C. R. Acad. Sci.
Paris Sér. A-B, 269, (1969), A181–A183.
P. Liardet, Transformations rationnelles et ensembles algébriques.
Thèse 3e cycle, Université de Provence, Faculté des Sciences (1970).
P. Liardet, Sur les transformations polynomiales et rationnelles. In
Séminaire de Théorie des Nombres, 1971–1972 (Univ. BordeauxI, Talence), Exp. No. 29. Lab. Théorie des Nombres, Centre Nat.
Recherche Sci., Talence (1972). p. 20.

[L25]

[L26]

[L27]

P. Liardet, Sur une conjecture de W. Narkiewicz. C. R. Acad. Sci.
Paris Sér. A-B, 274, (1972), A1836–A1838.
P. Liardet, Sur une conjecture de Serge Lang. C. R. Acad. Sci. Paris
Sér. A, 279, (1974), 435–437.
P. Liardet, Premiére thèse: Sur la stabilité rationnelle ou algébriqued’ensembles de nombres algébriques; Deuxiéme thèse: Difféomorphismes du tore : théorie classique et théorie générique. Thèsed’État: Sciences mathématiques, Université d’Aix-Marseille II, Faculté des Sciences. (1975).
P. Liardet, Sur une conjecture de Serge Lang. In Journées Arithmétiques de Bordeaux (Conference, Univ. Bordeaux, 1974). Soc. Math.
France, Paris (1975). pp. 187–210. Astérisque, Nos. 24–25.
K. K. Kubota and P. Liardet, Réfutation d’une conjecture de W.
Narkiewicz. C. R. Acad. Sci. Paris Sér. A-B, 282(22), (1976), Ai,
A1261–A1264.
P. Liardet, Résultats de stabilité algébrique. In Séminaire de Théorie
des Nombres, 1975–1976 (Univ. Bordeaux I, Talence), Exp. No. 24.
Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence(1976). p. 6.
P. Liardet, Stabilité algébrique et topologies hilbertiennes. In Séminaire Delange-Pisot-Poitou, 17e année (1975/76), Théorie des nombres: Fasc. 1, Exp. No. 8. Secrétariat Math., Paris (1977). p. 9.
P. Liardet, Répartition et ergodicité. In Séminaire Delange-PisotPoitou, 19e année: 1977/78, Théorie des nombres, Fasc. 1. Secrétariat Math., Paris (1978). pp. Exp. No. 10, 12.
P. Liardet, Discrépance sur le cercle (1979). Primaths 1, Univ. Marseille.
P. Liardet, Propriétés génériques de processus croisés. Compositio
Math., 39(4), (1981), 303–325.
J. Coquet and P. Liardet, Répartitions uniformes des suites et indépendance statistique. Compositio Math., 51(2), (1984), 215–236.
J. Coquet and P. Liardet, A metric study involving independent sequences. J. Analyse Math., 49, (1987), 15–53.
P. Liardet, Regularities of distribution. Compositio Math., 61(3),(1987), 267–293.
H. Jager and P. Liardet, Distributions arithmétiques des dénominateurs de convergents de fractions continues. Nederl. Akad. Wetensch. Indag. Math., 50(2), (1988), 181–197.
P. Liardet, Propriétés harmoniques de la numération suivant Jean
Coquet. In Colloque de Théorie Analytique des Nombres “Jean Coquet” (Marseille, 1985), vol. 88 of Publ. Math. Orsay. Univ. Paris
XI, Orsay (1988). pp. 1–35.
P. Liardet, Some metric properties of subsequences. Acta Arith.,
55(2), (1990), 119–135. ISSN 0065-1036.
J.-P. Allouche and P. Liardet, Generalized Rudin-Shapiro sequences.
Acta Arith., 60(1), (1991), 1–27.
P. Gabriel, M. Lemańczyk, and P. Liardet, Ensemble d’invariants
pour les produits croisés de Anzai. Mém. Soc. Math. France (N.S.),47, (1991), 1–102.
C. Kraaikamp and P. Liardet, Good approximations and continued
fractions. Proc. Amer. Math. Soc., 112(2), (1991), 303–309.
A. Broglio and P. Liardet, Predictions with automata. In Symbolic
dynamics and its applications (New Haven, CT, 1991), vol. 135 ofContemp. Math. Amer. Math. Soc., Providence, RI (1992). pp. 111–
124.
G. R. Goodson, J. Kwiatkowski, M. Lemańczyk, and P. Liardet, On
the multiplicity function of ergodic group extensions of rotations.
Studia Math., 102(2), (1992), 157–174.
M. Lemańczyk, P. Liardet, and J.-P. Thouvenot, Coalescence of circle extensions of measure-preserving transformations. Ergodic Theory Dynam. Systems, 12(4), (1992), 769–789.
S. Harari and P. Liardet, Rational interval maps and cryptography.
In Eurocode ’92 (Udine, 1992), vol. 339 of CISM Courses and Lectures. Springer, Vienna (1993). pp. 185–200.
P. Liardet, Automata and generalized Rudin-Shapiro sequences. Arbeitsberichte des Instituts für Mathematik der Universität Salzburg(1993).

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B. Kamiński and P. Liardet, Spectrum of multidimensional dynamical systems with positive entropy. Studia Math., 108(1), (1994),
77–85.
P. J. Grabner, P. Liardet, and R. F. Tichy, Odometers and systems ofnumeration. Acta Arith., 70, (1995), 103–123.
B. Kamiński, Z. S. Kowalski, and P. Liardet, On extremal and perfect σ-algebras for Zd -actions on a Lebesgue space. Studia Math.,
124(2), (1997), 173–178.
P. Liardet and D. Volný, Sums of continuous and differentiable functions in dynamical systems. Israel J. Math., 98, (1997), 29–60.
P. Liardet and P. Stambul, Algebraic computations with continued
fractions. J. Number Theory, 73(1), (1998), 92–121.
P. J. Grabner and P. Liardet, Harmonic properties of the sum-ofdigits function for complex bases. Acta Arith., 91, (1999), 329–349.
P. J. Grabner, P. Liardet, and R. F. Tichy, Average case analysis ofnumerical integration. In Advances in multivariate approximation(Witten-Bommerholz, 1998), vol. 107 of Math. Res. Wiley-VCH,
Berlin (1999). pp. 185–200.
G. Barat, T. Downarowicz, A. Iwanik, and P. Liardet, Propriétés
topologiques et combinatoires des échelles de numération. Colloquium Mathematicum, 84/85, (2000), 285–306.
Z. S. Kowalski and P. Liardet, Genericity of the K-property for aclass of transformations. Proc. Amer. Math. Soc., 128(10), (2000),
2981–2988.
P. Liardet and P. Stambul, Séries de Engel et fractions continuées. J.
Théor. Nombres Bordeaux, 12(1), (2000), 37–68.
U. Cerruti, M. Giacobini, and P. Liardet, Prediction of binary sequences by evolving finite state machines. In Proceedings of the Fifth
Conference on Artificial Evolution (EA-2001) (edited by P. Collet,
C. Fonlupt, J.-K. Hao, E. Lutton, and M. Schoenauer), vol. 2310 of
LNCS. Springer Verlag, Le Creusot, France, pp. 42–53.
I. Kátai and P. Liardet, Additive functions with respect to expansions
over the set of Gaussian integers. Acta Arith., 99(2), (2001), 173–
182.
G. Barat, T. Downarowicz, and P. Liardet, Dynamiques assocées àune échelle de numération. Acta Arith., 103, (2002), 41–77.
S. Aupetit, P. Liardet, and M. Slimane, Evolutionary search for binary strings with low aperiodic auto-correlations. Lecture Notes
in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2936, (2004),
39–50.
G. Barat and P. Liardet, Dynamical systems originated in the Ostrowski alpha-expansion. Ann. Univ. Sci. Budapest. Sect. Comput.,
24, (2004), 133–184.
P. J. Grabner, P. Liardet, and R. F. Tichy, Spectral disjointness ofdynamical systems related to some arithmetic functions. Publ. Math.
Debrecen, 66(1-2), (2005), 213–243.
S. Aupetit, N. Monmarché, M. Slimane, and P. Liardet, An exponential representation in the API algorithm for hidden Markov models
training. Lecture Notes in Computer Science (including subseries
Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 3871 LNCS, (2006), 61–72.
G. Barat, V. Berthé, P. Liardet, and J. Thuswaldner, Dynamical directions in numeration. Ann. Inst. Fourier (Grenoble), 56(7), (2006),
1987–2092. Numération, pavages, substitutions.
A. Bonnecaze, P. Liardet, A. Gabillon, and K. Blibech, Secure
time-stamping schemes: A distributed point of view. Annales
des Telecommunications/Annals of Telecommunications, 61(5-6),(2006), 662–681.
K. Dajani, C. Kraaikamp, and P. Liardet, Ergodic properties ofsigned binary expansions. Discrete Contin. Dyn. Syst., 15(1),(2006), 87–119.
F. M. Dekking and P. Liardet, On random walks in random scenery.
In Dynamics & stochastics, vol. 48 of IMS Lecture Notes Monogr.
Ser. Inst. Math. Statist., Beachwood, OH (2006). pp. 47–52.
P. Hellekalek and P. Liardet, The dynamics associated with certain
digital sequences. In Probability and number theory—Kanazawa
2005, vol. 49 of Adv. Stud. Pure Math. Math. Soc. Japan, Tokyo(2007). pp. 105–131.

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M. Drmota, P. J. Grabner, and P. Liardet, Block additive functions on
the Gaussian integers. Acta Arith., 135(4), (2008), 299–332.
M. Jebalia, A. Auger, and P. Liardet, Log-linear convergence andoptimal bounds for the (1 + 1)-ES. In Artificial evolution, vol. 4926
of Lecture Notes in Comput. Sci. Springer, Berlin (2008). pp. 207–
218.
I. Abou and P. Liardet, Flots chaînés. In Proceedings of the Sixth
Congress of Romanian Mathematicians. Vol. 1. Ed. Acad. Române,
Bucharest, pp. 401–432.
S. Aupetit, N. Monmarché, P. Liardet, and M. Slimane, Dissimilarity
analysis and application to visual comparisons. Studies in Computational Intelligence, 201, (2009), 333–361.
V. Baláž, P. Liardet, and O. Strauch, Distribution functions of thesequence ϕ(M)/M, M ∈ (k, k + N] as k, N go to infinity. Integers,
10, (2010), A53, 705–732.
A. Bonnecaze and P. Liardet, Efficient uniform k-out-of-n generators. In Proceedings - 5th International Conference on Systems andNetworks Communications, ICSNC 2010. IEEE Computer Society,pp. 177–182.
A. Bonnecaze and P. Liardet, Uniform generators and combinatorial designs. International Journal On Advances in Networks andServices, 4, (2011), 107–118.
P. J. Grabner, P. Hellekalek, and P. Liardet, The dynamical pointof view of low-discrepancy sequences. Unif. Distrib. Theory, 7(1),(2012), 11–70.
P. Liardet, Gérard Rauzy (1938–2010). Gaz. Math., 132, (2012),
92–99.
P. Liardet, Gérard Rauzy (1938–2010). Unif. Distrib. Theory, 7(1),(2012), 1–9.
I. Abou and P. Liardet, Mixing of Prouhet-Thue-Morse and RudinShapiro sequences. Ann. Univ. Sci. Budapest. Sect. Comput., 40,(2013), 55–67.
A. Bonnecaze, P. Liardet, and A. Venelli, AES side-channel countermeasure using random tower field constructions. Des. Codes Cryptogr., 69(3), (2013), 331–349.
D.-J. Feng, P. Liardet, and A. Thomas, Partition functions in numeration systems with bounded multiplicity. Unif. Distrib. Theory, 9(1),(2014), 43–77.

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