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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.
Reviewer: Reviewer (Berlin)

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