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

TL;DR

This paper extends the metric theory of small gaps from integer sequences to real sequences, providing bounds for floored minimal gaps and analyzing well-spaced sequences, building on prior work involving additive energy and difference sets.

Contribution

It generalizes key results from integer sequences to real sequences, offering new bounds for floored minimal gaps and broadening the class of sequences analyzed.

Findings

Established upper bounds for floored minimal gaps of real sequences.

Proved lower bounds for well-spaced and broader classes of sequences.

Recovered and extended previous theorems from integer sequence case.

Abstract

Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. The metric theory of minimal gaps for the sequence $\{\alpha a_n \text{ mod }1, 1\leq n \leq N\}$ 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 $\{a_1,\ldots,a_N\}$ 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 $\{a_1,\ldots,a_N\}$. 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 $\lfloor \delta^{\alpha}_{\min} \rfloor (N)$ for real sequences $(a_n)_{n \geq 1}$ 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.