The metric theory of small gaps for a sequence of real numbers

Abstract

Let $(a_n{)}_{n\ge 1}$ be a sequence of distinct positive integers. The metric theory of minimal gaps for the sequence $\{\alpha a_n\text{ mod }1,1\le n\le N\}$ as $N\to \infty$ was initiated by Rudnick, who established that the minimal gap admits an asymptotic upper bound expressible in terms of the additive energy of $\{a_1,\dots ,a_N\}$ for almost every $\alpha$. Later, Aistleitner, El-Baz, and Munsch demonstrated that the metric theory of minimal gaps for such sequences is governed not by the additive energy, but by the cardinality of the difference set of $\{a_1,\dots ,a_N\}$. They established a sharp convergence test for the typical asymptotic order of the minimal gap and proved general upper and lower bounds that are readily applicable. A key element of their proof relies on the resolution of the Duffin--Schaeffer conjecture by Koukoulopoulos and Maynard. In this article, we generalise several results from the article of Aistleitner, El-Baz, and Munsch on \emph{integer} sequences to the case of \emph{real} sequences. While an upper bound for ${\delta }_{\min }^{\alpha }(N)$ remains elusive, we obtain one for its floored counterpart $\lfloor {\delta }_{\min }^{\alpha }\rfloor (N)$ for real sequences $(a_n{)}_{n\ge 1}$ of distinct numbers. Our theorems recover Theorems 1-3, as well as the result from Section 4.3 of the article by Aistleitner, El-Baz, and Munsch. Furthermore, we establish lower bounds for the minimal gaps of well-spaced sequences and, more generally, of a broader family that contains them.

34 pages