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On almost perfect additive complements
Abstract
Let $A, B$ be infinite sets of non-negative integers with n∈ A+B for all sufficiently large integers n. We prove that if $A(x)B(x)-x$ does not tend to infinity, then limsupx→∞A(x)B(x)/x∈[3/2, 2]. The lower bound is sharp, and confirms a conjecture of Chen. Our proof is based on the weak limits and the convolution factorization theorem of Gabardo and Lai. A discrete proof of the same bound was also given by Zhang. Motivated by the known spectrum of perfect additive complements, we conjecture that the interval $(3/2, 8/5)$ is also excluded for almost perfect additive complements.
Related
BibTeX
@misc{doorn2026almost,
title = {{On almost perfect additive complements}},
author = {Wouter van Doorn and Boning Liu and Quanyu Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35027},
eprint = {2610.35027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35027},
}
RIS
TY - UNPB
TI - On almost perfect additive complements
AU - Wouter van Doorn
AU - Boning Liu
AU - Quanyu Tang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35027
UR - https://arxiv.org/abs/2610.35027
AB - Let $A, B$ be infinite sets of non-negative integers with n∈ A+B for all sufficiently large integers n. We prove that if $A(x)B(x)-x$ does not tend to infinity, then limsup_x→∞A(x)B(x)/x∈[3/2, 2]. The lower bound is sharp, and confirms a conjecture of Chen. Our proof is based on the weak limits and the convolution factorization theorem of Gabardo and Lai. A discrete proof of the same bound was also given by Zhang. Motivated by the known spectrum of perfect additive complements, we conjecture that the interval $(3/2, 8/5)$ is also excluded for almost perfect additive complements.
ER -