On a conjecture of Sárközy and Szemerédi
Volume 169 / 2015
Acta Arithmetica 169 (2015), 47-58
MSC: Primary 11B13, 11B34; Secondary 05A17.
DOI: 10.4064/aa169-1-3
Abstract
Two infinite sequences $A$ and $B$ of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additive complements $A$ and $B$ with $\limsup A(x)B(x)/x\le 1$ and $A(x)B(x)-x=O(\min\{ A(x),B(x)\})$, where $A(x)$ and $B(x)$ are the counting functions of $A$ and $B$, respectively. We prove that, for infinite additive complements $A$ and $B$, if $\limsup A(x)B(x)/x\le 1$, then, for any given $M>1$, we have $$A(x)B(x)-x\ge (\min \{ A(x), B(x)\})^M$$ for all sufficiently large integers $x$. This disproves the above Sárközy–Szemerédi conjecture. We also pose several problems for further research.
