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 | = n log 2 3 and that every subset A ′ ⊆ A of size k satisfies | A ′ − A ′ | ≥ k log 2 3 . This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.