Improved Bounds for Distinct Multiples in Intervals

Kaizhe Chen; Samuel Korsky

number theory

Improved Bounds for Distinct Multiples in Intervals

arXiv:2607.26450

summary

The paper establishes new lower and upper bounds for the functions F(n) and h_P(n), which measure the smallest interval length needed to contain distinct multiples of all integers (or primes) up to n, improving previous results and disproving a conjectured upper bound.

Abstract

For $n \geq 1$ , 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 $k ∣ a_{k}$ for $1 \leq k \leq n$ , and define $h_{P} (n)$ 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.

Improved lower and upper bounds

Topics & keywords

#distinct multiples #interval covering #lower bounds #upper bounds #combinatorial number theoryF(n)h_P(n)square‑residue digit constructionsum‑difference theoremexponential lower boundβ exponent