A metric theory of minimal gaps

TL;DR

This paper investigates the minimal gap statistics of fractional parts of sequences scaled by a real number, showing that under certain conditions, these gaps resemble those of a random sequence for almost all scaling factors.

Contribution

It introduces a new metric theory for minimal gaps in fractional sequences, connecting additive energy conditions to gap distribution behavior for almost all parameters.

Findings

Minimal gaps are close to those of random sequences under specific additive energy conditions.

The results hold for almost all scaling factors α.

Provides a theoretical framework linking additive properties to gap statistics.

Abstract

We study the minimal gap statistic for fractional parts of sequences of the form $\mathcal A^\alpha = \{\alpha a(n)\}$ where $\mathcal A = \{a(n)\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\alpha$, the minimal gap $\delta_{\min}^\alpha(N)=\min\{\alpha a(m)-\alpha a(n)\bmod 1: 1\leq m\neq n\leq N\}$ is close to that of a random sequence.