arXiv subjects

Pierre-Yves Bienvenu

Publications and source records attributed to Pierre-Yves Bienvenu.

18 recordsLinked to original sources

arXiv2026

A Generalization of Sárközy's theorem in function fields

Sárközy's theorem says that if $A \subset \mathbb{Z}$ has positive upper asymptotic density, then there are distinct $a_1, a_2 \in A$ and $n \in \mathbb{Z}$ such that $a_1-a_2 = n^2$. The same is true if $n^2$ is replaced by $F(n)$ for any polynomial $F \in \mathbb{Z}[x]$ with constant term zero. Green proved an $\mathbb{F}_q[t]$-analog of Sárközy's theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial $F \in \mathbb{F}_q[x]$. This condition was recently removed by Li and Sauermann. In this paper, we generalize Green's argument to accommodate equations in more variables in $\mathbb{F}_q[t]$, while pointing out that the technical condition can be removed by means of a simple observation.

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arXiv2026

Bohr sets in sumsets III: expanding difference sets and almost Bohr sets

Let $G$ be a discrete abelian group. Følner showed that if $A \subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \setminus E$ where $B$ is a Bohr set and $E$ has zero Banach density. We study the sets $S \subseteq G$ for which $A - A + S$ contains a Bohr set for every $A \subseteq G$ of positive upper Banach density. For $G = \mathbb{Z}$, we show that the sets $\{n^2: n \in \mathbb{N}\}$, $\{p - 1: p \text{ prime}\}$, and $\{ \lfloor n^c \rfloor: n \in \mathbb{N} \}$ with $c > 0$, have this property. Moreover, we prove that there are sets $A, B \subseteq \mathbb{Z}$ such that $A$ is dense in the Bohr topology of $\mathbb{Z}$, $d^*(B) > 0$, while $A + B$ is not piecewise Bohr, answering two questions of the second author in [31]. We also study those sets $S$ such that $A + S$ contains a Bohr set for every almost Bohr set $A$. As applications, we prove: (i) If $ϕ_1, ϕ_2: G \to G$ are (not necessarily commuting) homomorphisms with finite indices $[G: ϕ_i(G)]$, and $C \subseteq G$ is a central set, then $ϕ_1(C) - ϕ_1(C) + ϕ_2(C)$ contains a Bohr set. This answers one of our questions in [35] and generalizes results in [44, 48]; (ii) Every set of pointwise recurrence in $\mathbb{Z}$ is a set of nice recurrence and a van der Corput set, extending known properties of sets of pointwise recurrence studied in [26, 27, 40].

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arXiv2026

Additive index and Carlitz rank

We compare several complexity measures for self-mappings of finite fields. In particular, we show that Carlitz rank and additive index cannot be small simultaneously up to trivial exceptions. That is, these two measures detect cryptographic weaknesses of different classes of functions. We also study the relationship between additive index and degree or weight, respectively, complementing earlier results of Aksoy et al. and Gómez-Pérez et al. on the relationship between Carlitz rank and degree or weight, respectively. Finally, we show that a function closely related to the discrete logarithm provides an example in which all four complexity measures, degree, weight, additive index and Carlitz rank, are large.

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arXiv2026

On the additive index of the Diffie-Hellman mapping and the discrete logarithm

Several complexity measures such as degree, sparsity and multiplicative index for cryptographic functions including the Diffie-Hellman mapping and the discrete logarithm in a finite field have been studied in the literature. In 2022, Reis and Wang introduced another complexity measure, the additive index, of a self-mapping of a finite field. In this paper, under certain conditions, we determine lower bounds on the additive index of the univariate Diffie-Hellman mapping and a self-mapping of $\mathbb{F}_q$ which can be identified with the discrete logarithm in a finite field.

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arXiv2025

Realisability of simultaneous density constraints for sets of integers

In this note, we study the set $\mathcal{D}$ of values of the quadruplet $(\underline{\mathrm{d}}(A),\overline{\mathrm{d}}(A),\underline{\mathrm{d}}(2A),\overline{\mathrm{d}}(2A))$ where $A\subset\mathbb{N}$ and $\underline{\mathrm{d}},\overline{\mathrm{d}}$ denote the lower and upper asymptotic density, respectively. Completing existing results on the topic, we determine each of its six projections on coordinate planes, that is, the sets of possible values of the six subpairs of the quadruplet. Further, we show that this set $\mathcal{D}$ has non empty interior, in particular has positive measure. To do so, we use among others probabilistic and diophantine methods. Some auxiliary results pertaining to these methods may be of general interest.

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arXiv2024

On additive bases in infinite abelian semigroups

Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups and semigroups. We show that, for every infinite abelian group $T$, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $\mathbf{N}$. Also, using invariant means, we address a classical problem, initiated by Erdős and Graham and then generalized by Nash and Nathanson both in the case of $\mathbf{N}$, of estimating the maximal order $X_T(h,k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that $X_T(h,k)=O(h^{2k+1})$ for every integer $k \ge 1$. This result is new even in the case where $k=1$. Besides the maximal order $X_T(h,k)$, the typical order $S_T(h,k)$ is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in $\mathbf{N}$ and in abelian groups.

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arXiv2024

Intersective sets for sparse sets of integers

For $E \subset \mathbb{N}$, a subset $R \subset \mathbb{N}$ is $E$-intersective if for every $A \subset E$ having positive upper relative density, we have $R \cap (A - A) \neq \varnothing$. On the other hand, $R$ is chromatically $E$-intersective if for every finite partition $E=\bigcup_{i=1}^k E_i$, there exists $i$ such that $R\cap (E_i-E_i)\neq\varnothing$. When $E=\mathbb{N}$, we recover the usual notions of intersectivity and chromatic intersectivity. In this article, we investigate to which extent known intersectivity results hold in the relative setting when $E = \mathbb{P}$, the set of primes, or other sparse subsets of $\mathbb{N}$. Among other things, we prove: -There exists an intersective set that is not $\mathbb{P}$-intersective. -However, every $\mathbb{P}$-intersective set is intersective. -There exists a chromatically $\mathbb{P}$-intersective set which is not intersective (and therefore not $\mathbb{P}$-intersective). -The set of shifted Chen primes $\mathbb{P}_{\mathrm{Chen}} + 1$ is $\mathbb{P}$-intersective (and therefore intersective).

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arXiv2023

On algebraic properties of power monoids of numerical monoids

Let $S \subset \mathbb{N}_0$ be a numerical monoid and let $\mathcal P_{\mathrm{fin}} (S)$, resp $\mathcal P_{\mathrm{fin},0}(S)$, denote the power monoid, resp. the restricted power monoid, of $S$, that is the set of all finite nonempty subsets of $S$, resp. the set of all finite nonempty subsets of $S$ containing 0, with set addition as operation. The arithmetic of power monoids received some attention in recent literature. We complement these investigations by studying algebraic properties of power monoids, such as their prime spectrum. Moreover, we prove that almost all elements of $\mathcal P_{\mathrm{fin},0} (S)$ are irreducible (i.e., they are not proper sumsets), quantitatively improving a result of Shitov along the way.

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arXiv2022

A transference principle for systems of linear equations, and applications to almost twin primes

The transference principle of Green and Tao enabled various authors to transfer Szemerédi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide a transference principle which applies to general affine-linear configurations of finite complexity. We illustrate the broad applicability of our transference principle with the case of almost twin primes, by which we mean either Chen primes or "bounded gap primes", as well as with the case of primes of the form $x^2+y^2+1$. Thus, we show that in these sets of primes the existence of solutions to finite complexity systems of linear equations is determined by natural local conditions. These applications rely on a recent work of the last two authors on Bombieri-Vinogradov type estimates for nilsequences.

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arXiv2021

Kneser's Theorem in $σ$-finite Abelian groups

Let $G$ be a $σ$-finite abelian group, i.e. $G=\bigcup_{n\geq 1} G_n$ where $(G_n)_{n\geq 1}$ is a non decreasing sequence of finite subgroups. For any $A\subset G$, let $\underline{\mathrm{d}}(A):=\liminf_{n\to\infty}\frac{|A\cap G_n|}{|G_n|}$ be its lower asymptotic density. We show that for any subsets $A$ and $B$ of $G$, whenever $\underline{\mathrm{d}}(A+B)<\underline{\mathrm{d}}(A)+\underline{\mathrm{d}}(B)$, the sumset $A+B$ must be periodic, that is, a union of translates of a subgroup $H\leq G$ of finite index. This is exactly analogous to Kneser's theorem regarding the density of infinite sets of integers. Further, we show similar statements for the upper asymptotic density in the case where $A=\pm B$. An analagous statement had already been proven by Griesmer in the very general context of countable abelian groups, but the present paper provides a much simpler argument specifically tailored for the setting of $σ$-finite abelian groups. This argument relies on an appeal to another theorem of Kneser, namely the one regarding finite sumsets in an abelian group.

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arXiv2019

On the density or measure of sets and their sumsets in the integers or the circle

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leqα\leqβ\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that $\mathrm{d}(A)=α$ and $\mathrm{d}(A+A)=β$. More generally we study the set of $k$-tuples $(\mathrm{d}(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{N}$. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of ``pseudo $s$th powers''. We consider similar problems for subsets of the circle $\mathbb{R}/\mathbb{Z}$, that is, we partially determine the set of $k$-tuples $(μ(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{R}/\mathbb{Z}$.

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arXiv2018

A note on the set $\boldsymbol{A(A+A)}$

Let $p$ a large enough prime number. When $A$ is a subset of $\mathbb{F}_p\smallsetminus\{0\}$ of cardinality $|A|> (p+1)/3$, then an application of Cauchy-Davenport Theorem gives $\mathbb{F}_p\smallsetminus\{0\}\subset A(A+A)$. In this note, we improve on this and we show that if $|A|\ge 0.3051 p$ implies $A(A+A)\supseteq\mathbb{F}_p\smallsetminus\{0\}$. In the opposite direction we show that there exists a set $A$ such that $|A| > (1/8+o(1))p$ and $\mathbb{F}_p\smallsetminus\{0\}\not\subseteq A(A+A)$.

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arXiv2018

A note on the Bilinear Bogolyubov Theorem: Transverse and bilinear sets

A set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ is called $\textit{bilinear}$ when it is the zero set of a family of linear and bilinear forms, and $\textit{transverse}$ when it is stable under vertical and horizontal sums. A theorem of the first author provides a generalization of Bogolyubov's theorem to the bilinear setting. Roughly speaking, it implies that any dense transverse set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ contains a large bilinear set. In this paper, we elucidate the extent to which a transverse set is forced to be (and not only contain) a bilinear set.

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arXiv2018

Linear and quadratic uniformity of the Möbius function over $\mathbb{F}_q[t]$

We examine correlations of the Möbius function over $\mathbb{F}_q[t]$ with linear or quadratic phases, that is, averages of the form \begin{equation} \label{eq:average} \frac{1}{q^n}\sum_{\text{deg }f 0$ if $Q$ is linear and $O \left( q^{-n^c} \right)$ for some absolute constant $c>0$ if $Q$ is quadratic. The latter bound may be reduced to $O(q^{-c'n}$) for some $c'>0$ when $Q(f)$ is a linear form in the coefficients of $f^2$, that is, a Hankel quadratic form, whereas for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.

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arXiv2018

A bilinear Bogolyubov theorem

The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums and differences on $P$, that is, operations on the second coordinate when the first one is fixed, or vice versa. The structure we find is the zero set of a family of bilinear forms on a Cartesian product of vector subspaces. The codimensions of the subspaces and the number of bilinear forms involved are bounded by a function $c(δ)$ of the density $δ=\lvert P\rvert/p^{2n}$ only. The proof uses various tools of additive combinatorics, such as the (linear) Bogolyubov theorem, the density increment method, as well as the Balog-Szemerédi-Gowers and Freiman-Ruzsa theorems.

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arXiv2017

Polynomial equations in function fields

The breakthrough paper of Croot, Lev, Pach \cite{CLP} on progression-free sets in $\Z_4^n$ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem \cite{EG}. Using this method, we bound the size of a set of polynomials over $\F_q$ of degree less than $n$ that is free of solutions to the equation $\sum_{i=1}^k a_if_i^r=0$, where the coefficients $a_i$ are polynomials that sum to 0 and the number of variables satisfies $k\geq 2r^2+1$. The bound we obtain is of the form $q^{cn}$ for some constant $c<1$. This is in contrast to the best bounds known for the corresponding problem in the integers, which offer only a logarithmic saving, but work already with as few as $k\geq r^2+1$ variables.

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arXiv2016

Asymptotics for some polynomial patterns in the primes

We prove asymptotic formulae for sums of the form $$ \sum_{n\in\mathbb{Z}^d\cap K}\prod_{i=1}^tF_i(ψ_i(n)), $$ where $K$ is a convex body, each $F_i$ is either the von Mangoldt function or the representation function of a quadratic form, and $Ψ=(ψ_1,\ldots,ψ_t)$ is a system of linear forms of finite complexity. When all the functions $F_i$ are equal to the von Mangoldt function, we recover a result of Green and Tao, while when they are all representation functions of quadratic forms, we recover a result of Matthiesen. Our formulae imply asymptotics for some polynomial patterns in the primes. Specifically, they describe the asymptotic behaviour of the number of $k$-term arithmetic progressions of primes whose common difference is a sum of two squares. The article combines ingredients from the work of Green and Tao on linear equations in primes and that of Matthiesen on linear correlations amongst integers represented by a quadratic form. To make the von Mangoldt function compatible with the representation function of a quadratic form, we provide a new pseudorandom majorant for both -- an average of the known majorants for each of the functions -- and prove that it has the required pseudorandomness properties.

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arXiv2016

A higher-dimensional Siegel-Walfisz theorem

The Green-Tao-Ziegler theorem provides asymptotics for the number of prime tuples of the form $(ψ_1(n),\ldots,ψ_t(n))$ when $n$ ranges among the integer vectors of a convex body $K\subset [-N,N]^d$ and $Ψ=(ψ_1,\ldots,ψ_t)$ is a system of affine-linear forms whose linear coefficients remain bounded (in terms of $N$). In the $t=1$ case, the Siegel-Walfisz theorem shows that the asymptotic still holds when the coefficients vary like a power of $\log N$. We prove a higher-dimensional (i.e. $t>1$) version of this fact. We provide natural examples where our theorem goes beyond the one of Green and Tao, such as the count of arithmetic of progressions of step $\lfloor \log N\rfloor$ times a prime in the primes up to $N$. We also apply our theorem to the determination of asymptotics for the number of linear patterns in a dense subset of the primes, namely the primes $p$ for which $p-1$ is squarefree. To the best of our knowledge, this is the first such result in dense subsets of primes save for congruence classes.

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