Infinite Generalized Sidon Sets with a Negative Answer to a Problem of Szemerédi
Abstract
We construct infinite Bh sets that repeatedly attain the density scale of finite Bh sets. For every integer h≥2 and every finite Bh set F of positive integers, we obtain an infinite Bh set A⊇ F such that limsupx→∞A(x)/x1/h=1/h, liminfx→∞A(x)/x1/h=0. The key is to preserve density while preventing additive relations between successive blocks. A positional encoding of the finite-field moment curve has q elements in an interval of length Oh(qh), while, for each 1≤ r<h, sums arising from distinct r-element multisets are separated by more than q. This separation allows whole blocks to be appended to a prescribed finite set without a further dilation. In particular, our result gives a negative answer to a problem posed by Szemerédi \cite{szemeredi}.
Related
BibTeX
@misc{wang2026infinite,
title = {{Infinite Generalized Sidon Sets with a Negative Answer to a Problem of Szemerédi}},
author = {Guoqing Wang and Wenkai Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01871},
eprint = {2611.01871},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01871},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinite Generalized Sidon Sets with a Negative Answer to a Problem of Szemerédi
AU - Guoqing Wang
AU - Wenkai Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01871
UR - https://arxiv.org/abs/2611.01871
AB - We construct infinite B_h sets that repeatedly attain the density scale of finite B_h sets. For every integer h≥2 and every finite B_h set F of positive integers, we obtain an infinite B_h set A⊇ F such that limsup_x→∞A(x)/x^1/h=1/h, liminf_x→∞A(x)/x^1/h=0. The key is to preserve density while preventing additive relations between successive blocks. A positional encoding of the finite-field moment curve has q elements in an interval of length O_h(q^h), while, for each 1≤ r<h, sums arising from distinct r-element multisets are separated by more than q. This separation allows whole blocks to be appended to a prescribed finite set without a further dilation. In particular, our result gives a negative answer to a problem posed by Szemerédi \cite{szemeredi}.
ER -