On a conjecture of Sárközy and Szemerédi

Yong-Gao Chen. "On a conjecture of Sárközy and Szemerédi." Acta Arithmetica 169.1 (2015): 47-58. <http://eudml.org/doc/279289>.

@article{Yong2015,
abstract = {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 $A(x)B(x) - x ≥ (min\{A(x), B(x)\})^M$ for all sufficiently large integers x. This disproves the above Sárközy-Szemerédi conjecture. We also pose several problems for further research.},
author = {Yong-Gao Chen},
journal = {Acta Arithmetica},
keywords = {additive complements; sequences; counting functions},
language = {eng},
number = {1},
pages = {47-58},
title = {On a conjecture of Sárközy and Szemerédi},
url = {http://eudml.org/doc/279289},
volume = {169},
year = {2015},
}

TY - JOUR
AU - Yong-Gao Chen
TI - On a conjecture of Sárközy and Szemerédi
JO - Acta Arithmetica
PY - 2015
VL - 169
IS - 1
SP - 47
EP - 58
AB - 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 $A(x)B(x) - x ≥ (min{A(x), B(x)})^M$ for all sufficiently large integers x. This disproves the above Sárközy-Szemerédi conjecture. We also pose several problems for further research.
LA - eng
KW - additive complements; sequences; counting functions
UR - http://eudml.org/doc/279289
ER -