发表机构
Indian Institute of Science Education and Research Bhopal（印度科学教育研究所博帕尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明[N]^d的西顿子集是傅里叶一致的，结合高维Rado定理得出分块正则线性方程在其稠密子集有限分划下的单色解数量下界，还推导了其等分布性质，证明受Ortega与Prendiville的论证启发。
AI 中文摘要
本文证明了[N]^d的西顿子集是傅里叶一致的，并为西顿集证明了一个稠密模型引理。利用这些结果和高维版本的Rado定理，我们证明：对于任意含s≥5个非零系数变量的分块正则线性方程，以及[N]^d中足够稠密的西顿子集S的任意有限分划，当N足够大时，该方程的单色解数量满足≫|S|^s N^{-d}。作为傅里叶一致结果的推论，我们还证明[N]^d的稠密西顿子集在某些算术和Bohr结构中是等分布的。我们的证明受Ortega和Prendiville的论证启发。
英文摘要
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 $s \geq 5$ 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 $\gg |S|^s N^{-d}$ 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.
