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.

Disclosure

“|ΓI(m,H) (S)| ≥ |ΓIb (S)| ≥ Bc|S|1/θ ≥ n1−1/θ |S|1/θ ≥ |S|. b=1 Hall’s theorem implies F (n) ≤ H ≪η n2−1/θ . Letting η → 0 gives the desired statement. Statement on AI The authors used ChatGPT-5.6 Sol as an exploratory and proof-auditing tool during the development and refinement of the arguments. The authors have carefully checked all mathematical proofs and take responsibility for the manuscript. References [1] P. Erdős, So”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:13:25.962514Z

Structural counts