A Construction for Difference Sets with Local Properties

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