Minimal Gaps and Additive Energy in real-valued sequences

TL;DR

This paper investigates the minimal gap statistics of real-valued sequences scaled by a real parameter, linking it to the sequences' additive energy, and establishes results under the Lindelöf Hypothesis and unconditionally.

Contribution

It connects the minimal gap behavior of scaled real sequences to their additive energy, providing both conditional and unconditional results and extending previous work to real-valued sequences.

Findings

Conditional on Lindelöf Hypothesis, minimal gaps mimic random sequences for low additive energy.

Unconditional results support the connection between additive energy and minimal gaps.

Converse theorems relate large additive energy to deviations in minimal gap behavior.

Abstract

We study the minimal gap statistic for sequences of the form $\left( \alpha x_n \right)_{n = 1}^{\infty}$ where $\left( x_n \right)_{n = 1}^{\infty}$ is a sequence of real numbers, and its connection to the additive energy of $\left( x_n \right)_{n = 1}^{\infty}$. Inspired by a recent paper of Aistleitner, El-Baz and Munsch we show conditionally on the Lindel\"{o}f Hypothesis that if the additive energy is of lowest possible order then for almost all $\alpha$, the minimal gap $\delta_{\min}^{\alpha} (N) = \min \left\{ \alpha x_m - \alpha x_n \bmod \ 1 : 1 \leq m \neq n \leq N \right\}$ is close to that of a random sequence, a result Rudnick showed for integer-valued sequences. We also show unconditional results in this direction, as well as some converse theorems about sequences with large additive energy.