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(minA(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 for all sufficiently large integers x. This disproves the above Sárközy-Szemerédi conjecture. We also pose several problems for further research.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-1-3,
author = {Yong-Gao Chen},
title = {On a conjecture of S\'ark\"ozy and Szemer\'edi},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {47-58},
zbl = {06441796},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-1-3}
}
Yong-Gao Chen. On a conjecture of Sárközy and Szemerédi. Acta Arithmetica, Tome 168 (2015) pp. 47-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-1-3/