A construction for difference sets with local properties

Sara Fish, Ben Lund, Adam Sheffer

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<math><mi is="true">A</mi></math> of n<math><mi is="true">n</mi></math> real numbers such that |A−A|=nlog23<math><mrow is="true"><mo is="true">|</mo><mi is="true">A</mi><mo is="true">−</mo><mi is="true">A</mi><mo is="true">|</mo></mrow><mo is="true">=</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><msub is="true"><mrow is="true"><mo class="qopname" is="true">log</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mn is="true">3</mn></mrow></msup></math> and that every subset A′⊆A<math><msup is="true"><mrow is="true"><mi is="true">A</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo is="true">⊆</mo><mi is="true">A</mi></math> of size k<math><mi is="true">k</mi></math> satisfies |A′−A′|≥klog23<math><mrow is="true"><mo is="true">|</mo><msup is="true"><mrow is="true"><mi is="true">A</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo is="true">−</mo><msup is="true"><mrow is="true"><mi is="true">A</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo is="true">|</mo></mrow><mo is="true">≥</mo><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><msub is="true"><mrow is="true"><mo class="qopname" is="true">log</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mn is="true">3</mn></mrow></msup></math>. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.