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Acta Arithmetica

2015 | 169 | 1 | 47-58

On a conjecture of Sárközy and Szemerédi

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Abstrakty

EN
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 lim sup A(x)B(x)/x ≤ 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 lim sup A(x)B(x)/x ≤ 1, then, for any given M > 1, we have
$A(x)B(x) - x ≥ (min{A(x), B(x)})^M$
for all sufficiently large integers x. This disproves the above Sárközy-Szemerédi conjecture. We also pose several problems for further research.

47-58

wydano
2015

Twórcy

autor
• School of Mathematical Sciences, and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, P.R. China