arXiv subjects

Tony Haddad

Publications and source records attributed to Tony Haddad.

3 recordsLinked to original sources

arXiv2025

Poisson--Dirichlet approximation for counting integers with divisors in an interval

We give a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space. By combining this with Arratia's coupling, recently refined by Koukoulopoulos and the author, we obtain a general strategy to reduce the problem of finding an asymptotic formula for the number of integers whose prime factorization lies in any given subset of $\ell^1(\mathbb R)$ to bounding two key probabilities measuring proximity to the boundary of the subset in question. We apply this strategy to obtain an asymptotic formula for counting integers in $[1, x]$ that have a divisor in an interval $(y, z)$ in the regime $z/y \to \infty$ as $x \to \infty$.

math.NT↗

arXiv2024

Visiting early at prime times

Given an integer $m \geq 2$ and a sufficiently large $q$, we apply a variant of the Maynard--Tao sieve weight to establish the existence of an arithmetic progression with common difference $q$ for which the $m$-th least prime in such progression is $\ll_m q$, which is best possible. As we vary over progressions instead of fixing a particular one, the nature of our result differs from others in the literature. Furthermore, we generalize our result to dynamical systems. The quality of the result depends crucially on the first return time, which we illustrate in the case of Diophantine approximation.

math.NT↗

arXiv2024

On Arratia's coupling and the Dirichlet law for the factors of a random integer

Let $x \ge 2$, let $N_x$ be an integer chosen uniformly at random from the set $\mathbb Z \cap [1, x]$, and let $(V_1, V_2, \ldots)$ be a Poisson--Dirichlet process of parameter $1$. We prove that there exists a coupling of these two random objects such that $$ \mathbb E \, \sum_{i \ge 1} |\log P_i- V_i\log x| \asymp 1, $$ where the implied constants are absolute and $N_x = P_1P_2 \cdots$ is the unique factorization of $N_x$ into primes or ones with the $P_i$'s being non-increasing. This establishes a 2002 conjecture of Arratia arXiv:1305.0941 who constructed a coupling for which the left-hand side in the above estimate is $\ll \log\!\log x$, and who also proved that the left-hand side is $\ge 1-o(1)$ for all couplings. In addition, we use our refined coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into $k$ parts proved in 2023 by Leung arXiv:2206.14728 and we improve on its error term.

math.NT↗