SUMSETS AND DIFFERENCE SETS CONTAINING A COMMON TERM OF A SEQUENCE

Source abstract

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/(2 β )) a 1 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 { 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 $\vert A\vert \gt \frac 45 n$ , then ( A + A )∩( A − A ) contains a power of 2. Furthermore, $\frac 45$ cannot be improved.