The metric theory of small gaps for a sequence of real numbers
Jewel Mahajan
TL;DR
This work extends the metric theory of minimal gaps for sequences $(\alpha a_n)$ modulo $1$ from the integer setting to arbitrary real sequences, revealing that the typical minimal gap is governed by the arithmetic structure of the floored difference set rather than additive energy. By introducing natural real-sequence variants of the gap and leveraging the resolution of the Duffin–Schaeffer conjecture, it establishes sharp, broadly applicable lower and upper bounds expressed in terms of $D_N$ and $H_N$, and proves a comprehensive zero–one law for the floored gap. The results recover and generalise Rudnick and ABM in the real setting and yield concrete bounds for well-studied families such as lacunary, Beatty, polynomial, and smooth-image sequences. The methodology combines $L^2$-type arguments with gcd-sum estimates and DS machinery to handle overlaps, yielding both infinitely often and all-large-$N$ bounds, and providing a unified framework for the local statistics of real sequences modulo one. These findings deepen the connection between Diophantine approximation and the fine-scale distribution of deterministic sequences, with potential applications to lacunary and Beatty-type systems and to sequences arising from smooth functions.
Abstract
Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. The metric theory of minimal gaps for the sequence $\{αa_n \text{ mod }1, 1\leq n \leq N\}$ 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 $α$. 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 $δ_{\min}^α(N)$ remains elusive, we obtain one for its floored counterpart $\lfloor δ^α_{\min} \rfloor (N)$ 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.