A Construction for Difference Sets with Local Properties

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 $A^{\prime }\subseteq A$ of size $k$ satisfies $|A^{\prime }-A^{\prime }|\ge k^{{\log }_{2}3}$. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.