On a conjecture of Sárközy and Szemerédi

Yong-Gao Chen ¹   ; Jin-Hui Fang ²

¹ School of Mathematical Sciences and Institute of Mathematics Nanjing Normal University Nanjing 210023, P.R. China
² Department of Mathematics Nanjing University of Information Science & Technology Nanjing 210044, P.R. China

Acta Arithmetica, Tome 169 (2015) no. 1, pp. 47-58

Résumé

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.

Détail

DOI : 10.4064/aa169-1-3

Mots-clés : infinite sequences non negative integers called infinite additive complements their sum contains sufficiently large integers szemer conjectured there exist infinite additive complements limsup x x x min where counting functions respectively prove infinite additive complements limsup x given have x x min sufficiently large integers disproves above szemer conjecture pose several problems further research