arXiv subjects

Cédric Pilatte

Publications and source records attributed to Cédric Pilatte.

8 recordsLinked to original sources

arXiv2026

Improved bounds for the two-point logarithmic Chowla conjecture

Let $λ$ be the Liouville function, defined as $λ(n) := (-1)^{Ω(n)}$ where $Ω(n)$ is the number of prime factors of $n$ with multiplicity. In 2021, Helfgott and Radziwiłł proved that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll \frac{\log x}{(\log \log x)^{1/2}},$$improving earlier results by Tao and Teräväinen. We prove that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll (\log x)^{1-c}$$for some absolute constant $c>0$. This appears to be best possible with current methods.

math.NT↗

arXiv2026

Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(\mathbb{Z}/N\mathbb{Z})^{\times}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.

math.NT↗

arXiv2026

Improved bounds for the Fourier uniformity conjecture

Let $λ$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{α\in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} λ(n) e(nα)\bigg\rvert = o(HX)$$ as $X\to \infty$, in the regime $H = H(X) \geq \exp((\log X)^{2/5+\varepsilon})$. This improves upon a result of Walsh towards the Fourier uniformity conjecture.

math.NT↗

arXiv2023

A solution to the Erdős-Sárközy-Sós problem on asymptotic Sidon bases of order 3

A set $S\subset \mathbb{N}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leq s_2$) are distinct. A set $S\subset \mathbb{N}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb{F}_q[t]$-analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.

math.NT↗

arXiv2022

New bound for Roth's theorem with generalized coefficients

We prove the following conjecture of Shkredov and Solymosi: every subset $A \subset \mathbf{Z}^2$ such that $\sum_{a\in A\setminus\{0\}} 1/\left\|a\right\|^{2} = +\infty$ contains the three vertices of an isosceles right triangle. To do this, we adapt the proof of the recent breakthrough by Bloom and Sisask on sets without three-term arithmetic progressions, to handle more general equations of the form $T_1a_1+T_2a_2+T_3a_3 = 0$ in a finite abelian group $G$, where the $T_i$'s are automorphisms of $G$.

math.CO↗

arXiv2021

A note on optimal degree-three spanners of the square lattice

In this short note, we prove that the degree-three dilation of the square lattice $\mathbb{Z}^2$ is $1+\sqrt{2}$. This disproves a conjecture of Dumitrescu and Ghosh. We give a computer-assisted proof of a local-global property for the uncountable set of geometric graphs achieving the optimal dilation.

cs.CG↗

arXiv2021

On the sets of $n$ points forming $n+1$ directions

Let $S$ be a set of $n\geq 7$ points in the plane, no three of which are collinear. Suppose that $S$ determines $n+1$ directions. That is to say, the segments whose endpoints are in $S$ form $n+1$ distinct slopes. We prove that $S$ is, up to an affine transformation, equal to $n$ of the vertices of a regular $(n+1)$-gon. This result was conjectured in 1986 by R. E. Jamison. In an addendum to the paper, we show that a much stronger result can be obtained as a corollary of a structure theorem of Green and Tao on point sets spanning few ordinary lines.

math.CO↗

arXiv2020

NP-completeness of slope-constrained drawing of complete graphs

We prove the NP-completeness of the following problem. Given a set $S$ of $n$ slopes and an integer $k\geq 1$, is it possible to draw a complete graph on $k$ vertices in the plane using only slopes from $S$? Equivalently, does there exist a set $K$ of $k$ points in general position such that the slope of every segment between two points of $K$ is in $S$? We then present a polynomial algorithm for this question when $n\leq 2k-c$, conditional on a conjecture of R.E. Jamison. For $n=k$, an algorithm in $\mathcal{O}(n^4)$ was proposed by Wade and Chu. For this case, our algorithm is linear and does not rely on Jamison's conjecture.

cs.CG↗