SUMSETS AND DIFFERENCE SETS CONTAINING A COMMON TERM OF A SEQUENCE

Abstract

AbstractLet β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/ak≤β for all k≥1. The following result is proved: if n is an integer with n>(1+1/(2β))a1 and A is a subset of {0,1,…,n} with $\vert A\vert \ge (1- 1/({2\beta +1})) n +\frac 12$, then (A+A)∩(A−A) contains a term of {ak }. 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 $\vert A\vert \gt \frac 45 n$, then (A+A)∩(A−A) contains a power of 2. Furthermore, $\frac 45$ cannot be improved.