Infinite Sidon-type sets for zero-sum linear forms
Abstract
Let h ≥ 2, and let b = (b1,…,bh)∈ ℤh be a zero-sum vector with nonzero coordinates. For a set A={a1<a2<⋯}⊆ℕ, let rA,b(n) denote the number of h-tuples (x1,…,xh) of pairwise distinct elements of A satisfying b1x1+⋯+bhxh=n. We study density restrictions on sets A for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case b = (c1,−c1,…,ck,−ck), we prove that if A(x)/(x/log x)1/2k→∞, then 1/x∑|n|≤ x rA,b(n)→∞, whereas if A(x)≫ x1/2k, then 1/x∑|n|≤ x rA,b(n)≫log x. This recovers Chen's theorem on B2k-sequences. For general zero-sum vectors b, we prove analogous bounds under gap conditions: if an+1−an=o(nh−1log n), then 1/x∑|n|≤ x rA,b(n)→∞, whereas if an+1−an≪ nh−1, then 1/x∑|n|≤ x rA,b(n)≫log x.
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BibTeX
@misc{tafula2026infinite,
title = {{Infinite Sidon-type sets for zero-sum linear forms}},
author = {Christian Táfula},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.20753},
eprint = {2607.20753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20753},
}
RIS
TY - UNPB
TI - Infinite Sidon-type sets for zero-sum linear forms
AU - Christian Táfula
PY - 2026
DA - 2026/07/22
PB - arXiv
DO - 10.48550/arxiv.2607.20753
UR - https://arxiv.org/abs/2607.20753
AB - Let h ≥ 2, and let b = (b_1,…,b_h)∈ ℤ^h be a zero-sum vector with nonzero coordinates. For a set A={a_1<a_2<⋯}⊆ℕ, let r_A,b(n) denote the number of h-tuples (x_1,…,x_h) of pairwise distinct elements of A satisfying b_1x_1+⋯+b_hx_h=n. We study density restrictions on sets A for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case b = (c_1,−c_1,…,c_k,−c_k), we prove that if A(x)/(x/log x)^1/2k→∞, then 1/x∑_|n|≤ x r_A,b(n)→∞, whereas if A(x)≫ x^1/2k, then 1/x∑_|n|≤ x r_A,b(n)≫log x. This recovers Chen's theorem on B_2k-sequences. For general zero-sum vectors b, we prove analogous bounds under gap conditions: if a_n+1−a_n=o(n^h−1log n), then 1/x∑_|n|≤ x r_A,b(n)→∞, whereas if a_n+1−a_n≪ n^h−1, then 1/x∑_|n|≤ x r_A,b(n)≫log x.
ER -