Gaps between fractional parts, and additive combinatorics

TL;DR

This paper investigates the structure of finite sets by analyzing the differences between elements and their nearest neighbors, providing bounds on the number of distinct such differences to deepen understanding in additive combinatorics.

Contribution

It introduces new bounds on the number of distinct differences between elements and their nearest neighbors in finite sets, advancing additive combinatorics theory.

Findings

Bounds on the number of distinct differences $N_a - a$ for finite sets.

Insights into the structure of finite sets related to additive properties.

Enhanced understanding of the relationship between elements and their nearest neighbors.

Abstract

We give bounds on the number of distinct differences $N_a-a$ as $a$ varies over all elements of a given finite set $A$, and $N_a$ is a nearest neighbour to $a$.