On a conjecture of Sárközy and Szemerédi

• Autorzy: Yong-Gao Chen
• Języki publikacji: EN

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.

Kategorie tematyczne

• 11B34: Representation functions
• 05A17: Partitions of integers
• 11B13: Additive bases, including sumsets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 169
• Numer: 1
• Strony: 47-58

Daty

wydano

2015

Twórcy

autor

Yong-Gao Chen

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

Identyfikatory

DOI

10.4064/aa169-1-3