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Partition regular linear equations over Sidon sets
Abstract
In this article, we show that the Sidon subsets of [N]d are Fourier uniform, and we prove a dense model lemma for Sidon sets. Using these results and a higher-dimensional version of Rado's theorem, we prove that given any partition regular linear equation in s ≥ 5 variables with nonzero coefficients, and any finite partition of a sufficiently dense Sidon subset S of [N]d, the number of monochromatic solutions to this equation is ≫ |S|s N−d for all large N. As a corollary of the Fourier uniformity result, we show that dense Sidon subsets of [N]d are equidistributed in certain arithmetic and Bohr structures. Our proofs are motivated by the arguments of Ortega and Prendiville.
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BibTeX
@misc{pandey2026partition,
title = {{Partition regular linear equations over Sidon sets}},
author = {Dev Ranjan Pandey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11840},
eprint = {2610.11840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11840},
}
RIS
TY - UNPB
TI - Partition regular linear equations over Sidon sets
AU - Dev Ranjan Pandey
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11840
UR - https://arxiv.org/abs/2610.11840
AB - In this article, we show that the Sidon subsets of [N]^d are Fourier uniform, and we prove a dense model lemma for Sidon sets. Using these results and a higher-dimensional version of Rado's theorem, we prove that given any partition regular linear equation in s ≥ 5 variables with nonzero coefficients, and any finite partition of a sufficiently dense Sidon subset S of [N]^d, the number of monochromatic solutions to this equation is ≫ |S|^s N^−d for all large N. As a corollary of the Fourier uniformity result, we show that dense Sidon subsets of [N]^d are equidistributed in certain arithmetic and Bohr structures. Our proofs are motivated by the arguments of Ortega and Prendiville.
ER -