Searcharxiv⌕ Search

arXiv subjects

François Hennecart

Publications and source records attributed to François Hennecart.

9 recordsLinked to original sources
arXiv2024

About Berge-Füredi's conjecture on the chromatic index of hypergraphs

We show that the chromatic index of a hypergraph $\mathcal{H}$ satisfies Berge-Füredi conjectured bound $\mathrm{q}(\mathcal{H})\le Δ([\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\mathrm{ar}(\mathcal{H})$ or on the maximum degree $Δ(\mathcal{H})$. This provides sharp information in connection with Erdős-Faber-Lovász Conjecture which deals with the coloring of a family of cliques that intersect pairwise in at most one vertex.

math.CO↗
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.

math.NT↗
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 $A\subset \mathbb{R}/\mathbb{Z}$.

math.NT↗
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 $\mathbb{F}_p\smallsetminus\{0\}\not\subseteq A(A+A)$.

math.NT↗
arXiv2013

Conditional expanding bounds for two-variables functions over prime fields

In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\bFp^*$ is the existenceness of $Δ(α)>0$ such that if $|A|\asymp p^α$ then $$ \max(|A+A|,|A\cdot A|)\gg |A|^{1+Δ(α)}, $$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$: if $|A|\asymp|B|\asymp p^α$ then $$ \max(|f(A,B)|,|g(A,B)|)\gg |A|^{1+Δ(α)} $$ for some $Δ(α)>0$.

math.NT↗
arXiv2013

A Structure result for bricks in Heisenberg groups

We show that for a sufficiently big \textit{brick} $B$ of the $(2n+1)$-dimensional Heisenberg group $H_n$ over the finite field $\mathbb{F}_p$, the product set $B\cdot B$ contains at least $|B|/p$ many cosets of some non trivial subgroup of $H_n$.

math.NT↗
arXiv2012

A note on Freiman models in Heisenberg groups

Green and Ruzsa recently proved that for any $s\ge2$, any small squaring set $A$ in a (multiplicative) abelian group, i.e. $|A\cdot A|<K|A|$, has a Freiman $s$-model: it means that there exists a group $G$ and a Freiman $s$-isomorphism from $A$ into $G$ such that $|G|<f(s,K)|A|$. In an unpublished note, Green proved that such a result does not necessarily hold in non abelian groups if $s\ge64$. The aim of this paper is improve Green's result by showing that it remains true under the weaker assumption $s\ge6$.

math.NT↗