arXiv subjects

Alexandre Bailleul

Publications and source records attributed to Alexandre Bailleul.

6 recordsLinked to original sources

arXiv2026

On the largest Sidon subset in a finite subset of $\mathbb{R}^N$

We obtain a new lower bound on the largest Sidon subset of an arbitrary finite set of integers. If $H(n)$ denotes the minimum, over all $n$-element subsets of $\mathbb Z$, of the largest Sidon subset they contain, we prove that $H(n) \geqslant \left(\frac{1}{3\sqrt 3}+o(1)\right)\sqrt n \gtrsim 0.19\sqrt n$. This improves a lower bound of Abbott related to a conjecture of Erd\H{o}s on Sidon subsets of arbitrary sets of integers. The main ingredient is a compression lemma which produces, from any finite set of integers, a large subset admitting an injective Freiman $2$-morphism into a cyclic group. Combined with Singer's covering of $\mathbb Z/(q^2+q+1)\mathbb Z$ by Sidon sets, this yields the stated bound. We further extend the result to finite subsets of $\mathbb R^N$, uniformly in the dimension, by means of a projection argument and a Dirichlet approximation preserving Sidon's equation. As a consequence, every set of $n$ points in $\mathbb R^N$ contains a Sidon subset of cardinality at least $\left(\frac{1}{3\sqrt 3}+o(1)\right)\sqrt n$. We also discuss an adaptation to $B_2[g]$ sets, obtaining a lower bound of order $\frac{1}{3\sqrt 3}\sqrt{gn}$, and explain how the method can be adapted to other linear additive constraints.

math.CO↗

arXiv2026

A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races

We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences of highly composite moduli $q$ such that those Skewes' numbers grow very rapidly in some sense. This disproves unconditionally a conjecture of Fiorilli. In the other direction, assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes' numbers. Our approach relies on a quantitative Kronecker-Weyl theorem formulated in terms of the $1$-Wasserstein metric to obtain explicit rates for the convergence to the limiting distributions in these races.

math.NT↗

arXiv2025

Large subsets avoiding algebraic patterns

We prove the existence of a subset of the torus with large sumsets and avoiding all linear patterns. This extends a result of K\"orner, who had shown that for any integer $q \geq 1$, there exists a subset $K$ of $\mathbb R/\mathbb Z$ satisfying no non-trivial linear relations of order $2q-1$ and such that $q.K$ has positive Lebesgue measure. Our method is based on transfinite induction, which also allows us to produce large sets in different senses (cardinality, outer Lebesgue measure or Hausdorff dimension) avoiding families of algebraic patterns, for example Sidon sets in infinite abelian groups with small $2$ and $3$-torsion or sets with no repeated distances in $\mathbb R^n$. We also discuss questions of measurability of such sets and the role of the axiom of choice in our constructions.

math.CO↗

arXiv2023

Exceptional biases in counting primes over functions fields

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

math.NT↗

arXiv2020

Explicit Kronecker-Weyl theorems and applications to prime number races

We prove explicit versions of the Kronecker-Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis. As an application, we propose an alternative approach to problems concerning asymptotic densities in prime number races, over number fields and over function fields in one variable over finite fields, in the language of random variables. Our approach allows us to prove new results on the existence and positivity of some of those densities, which, in the case of races over function fields, do not require any linear independence hypothesis.

math.NT↗

arXiv2020

Chebyshev's bias in dihedral and generalized quaternion Galois groups

We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral $D_{2n}$ or (generalized) quaternion $\mathbb H_{2n}$ of two-power order. In the spirit of recent work of Fiorilli and Jouve arXiv:2001.05428, we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical aspect, where the degree goes to infinity. Our main contribution uncovers in families of extensions a phenomenon, for which Ng gave numerical evidence : real zeros of Artin L-functions sometimes have a radical influence on the distribution of Frobenius elements.

math.NT↗