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# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## On a conjecture of Sárközy and Szemerédi

### Tom 169 / 2015

Acta Arithmetica 169 (2015), 47-58 MSC: Primary 11B13, 11B34; Secondary 05A17. DOI: 10.4064/aa169-1-3

#### Streszczenie

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.

#### Autorzy

• Yong-Gao ChenSchool of Mathematical Sciences
and Institute of Mathematics
Nanjing Normal University
Nanjing 210023, P.R. China
e-mail
• Jin-Hui FangDepartment of Mathematics
Nanjing University of
Information Science & Technology
Nanjing 210044, P.R. China
e-mail

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