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 |A−A|=nlog23 and that every subset A′⊆A of size k satisfies |A′−A′|≥klog23. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.