Sumsets and Difference Sets Containing a Common Term of a Sequence

Abstract

Let β > 1 be a real number, and let {a k } be an unbounded sequence of positive integers such that a k+1 /a k ≤ β for all k ≥ 1. The following result is proved: if n is an integer with n > (1 + 1/(2β))a 1 and A is a subset of {0, 1, . . . , n} with |A| ≥ (1 − 1/(2β + 1))n + 1 2 , then (A + A) ∩ (A − A) contains a term of {a k }. The lower bound for |A| is optimal. Beyond these, we also prove that if n ≥ 3 is an integer and A is a subset of {0, 1, . . . , n} with |A| >