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On a conjecture of Sárközy and Szemerédi. (English) Zbl 1370.11017
Summary: 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 E. Szemerédi [Acta Math. Hung. 64, No. 3, 237–245 (1994; Zbl 0816.11013)] conjectured that there exist infinite additive complements $$A$$ and $$B$$ with $$\limsup A(x)B(x)/x\leq 1$$ 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 $$\limsup A(x)B(x)/x\leq 1$$, then, for any given $$M>1$$, we have $A(x)B(x)-x\geq (\min \{ A(x), B(x)\})^M$ for all sufficiently large integers $$x$$. This disproves the above Sárközy-Szemerédi conjecture. We also pose several problems for further research.

##### MSC:
 11B13 Additive bases, including sumsets 11B34 Representation functions