# A Construction for Difference Sets with Local Properties

## TL;DR

This paper presents a novel construction of finite real number sets with small difference sets and strong local properties, providing new bounds for the problem of distinct distances with local constraints.

## Contribution

It introduces a new method to construct sets with specific difference set sizes and local properties, advancing the understanding of difference sets and distance problems.

## Key findings

- Constructed sets of size n with difference set size n^{log_2 3}
- Proved subsets of size k have difference sets at least k^{log_2 3}
- Established the first non-trivial upper bounds for local distance problems

## Abstract

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size $k$ satisfies $|A'-A'|\ge k^{\log_2 3}$. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.