Improved Bounds for Distinct Multiples in Intervals

In this note, we study two functions introduced by Erd\H{o}s and Pomerance. For any positive integer $n$, let $F(n)$ be the smallest integer $F>0$ such that any $F$ consecutive integers contain a distinct multiple for each positive integer at most $n$, and let $h_{\mathbb{P}}(n)$ be the smallest integer $H>0$ such that any $H$ consecutive integers contain a distinct multiple for each prime at most $n$. Based on the square-residue digit construction of Green and Ruzsa, we prove $$F(n)\ge h_{\mathbb P}(n)\ge n\exp\!\left(\frac{1}{50}\frac{\log n}{\log\log n}\right),$$ for sufficiently large $n$. This improves the previous bounds $h_{\mathbb P}(n)/n\to\infty$ by Ruzsa, $F(n)\gg n\log n/\log \log n$ by van Doorn, and $F(n)\gg n\log n$ by Kominers and, in particular, disproves the conjecture $F(n)\ll n\log n$ by Kominers. Moreover, we prove $$F(n)\le n^{\beta+o(1)}\ll n^{1.4031} \qquad {\rm and}\qquad h_{\mathbb P}(n) \ll \frac{n^{7/5}}{(\log n)^{2/5}},$$ where $\beta\in (1,2)$ is the root of $2\beta^3-8\beta^2+8\beta-1=0$. This improves the previous best bounds $F(n)\ll n^{3/2}$ and $h_{\mathbb{P}}(n)\ll n^{3/2}/\sqrt{\log n}$ by Erd\H{o}s and Pomerance in 1980. Our upper bound on $F(n)$ is a corollary of the sum--difference theorem of Katz and Tao, while the upper bound on $h_{\mathbb P}(n)$ is achieved via a novel combinatorial method.