Difference sets and the metric theory of small gaps

TL;DR

This paper investigates the metric theory of small gaps in sequences of positive integers modulo 1, revealing that the difference set size, not additive energy, governs the minimal gap behavior for almost all real numbers.

Contribution

It introduces a new perspective that the difference set size, rather than additive energy, controls the minimal gaps in sequences modulo 1, providing sharp criteria and bounds.

Findings

Difference set size influences minimal gaps more than additive energy.

Established a sharp convergence/divergence test for minimal gap order.

Applied results to sequences like primes, squares, and eigenvalue spectra.

Abstract

Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \bmod 1, 1 \leq n \leq N\}$ as $N \to \infty$, valid for Lebesgue-almost all $\alpha$ and formulated in terms of the additive energy of $\{a_1, \dots, a_N\}$. In the present paper we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of $\{a_1, \dots, a_N\}$. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap, and prove (slightly weaker) general upper and lower bounds which allow for a direct application. A major input for these results comes from the recent proof of the Duffin--Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick and Radziwill on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.