Intersections of sets of distances

TL;DR

This paper explores conditions under which sets of distances from different sets of natural numbers intersect and demonstrates a recurrence property for certain infinite sets with controlled growth.

Contribution

It establishes new conditions for the intersection of distance sets and extends Khintchine's Recurrence Theorem to specific classes of infinite sets.

Findings

Conditions for nonempty intersection of distance sets.

A variant of Khintchine's Recurrence Theorem for sets with $a_n \\ll n^{3/2}$.

Results apply to large classes of zero density sets.

Abstract

We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $\Delta(A)\cap\Delta(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero density sets. We also show that a variant of Khintchine's Recurrence Theorem holds for all infinite sets $A=\{a_1<a_2<...\}$ with $a_n\ll n^{3/2}$.