Sumsets and difference sets containing a common term of

Authors

  • Quan-Hui Yang
  • Yong-Gao Chen

Keywords:

sumsets, difference sets, sequences

Abstract

Let $\beta > 1$ be a real number, and let $\{ a_k\}$ be an unbounded sequence of positive integers such that $a_{k+1}/a_k\le \beta $ for all $k\ge 1$. The following result is proved: If $n$ is an integer with $n>(1+1/(2\beta ))a_1$ and $A$ is a subset of $\{ 0, 1, \ldots , n\}$ with $|A|\ge \left( 1-\frac 1{2\beta +1}\right) n +\frac 12,$ then $(A+A)\cap (A-A)$ contains a term of $\{ a_k\}$. The lower bound for $|A|$ is optimal. Beyond these, we also prove that if $n\ge 3$ is an integer and $A$ is a subset of $\{ 0, 1, \ldots , n\}$ with $|A|>\frac 45 n$, then $(A+A)\cap (A-A)$ contains a power of 2. Furthermore, $\frac 45$ cannot be improved. DOI: 10.1017/S0004972711002747

Published

2011-12-09

Issue

Vol. 85 No. 1 (2012)

Section

Articles
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