Improved Bounds for Distinct Multiples in Intervals

Abstract

For $n\ge1$, let $F(n)$ be the least $H$ such that any $H$ consecutive integers contain $n$ pairwise distinct integers $a_1, a_2, \dots, a_n$ with $k \mid a_k$ for $1\le k\le n$, and define $h_{\mathbb P}(n)$ analogously for the primes at most $n$. We prove \[ F(n)\le n^{4/3}\exp\!\left(O\!\left(\frac{\log n}{\log\log n}\right)\right), \qquad h_{\mathbb P}(n)\ll \frac{n^{4/3}}{(\log n)^{1/3}}, \] and \[ F(n) \ge h_{\mathbb P}(n)\ge n\exp\!\left( \left(\frac{\log 2}{2}-o(1)\right) \frac{\log n}{\log\log n} \right). \] The upper bounds follow from a new estimate for unions of arithmetic progressions. The lower bound adapts a quadratic-residue compression construction of Green and Ruzsa.