{"title":"Discrepancy of arithmetic progressions in grids","authors":"Jacob Fox, Max Wenqiang Xu, Yunkun Zhou","doi":"10.1112/mtk.12237","DOIUrl":null,"url":null,"abstract":"<p>We prove that the discrepancy of arithmetic progressions in the <i>d</i>-dimensional grid <math>\n <semantics>\n <msup>\n <mrow>\n <mo>{</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>⋯</mi>\n <mo>,</mo>\n <mi>N</mi>\n <mo>}</mo>\n </mrow>\n <mi>d</mi>\n </msup>\n <annotation>$\\lbrace 1, \\dots, N\\rbrace ^d$</annotation>\n </semantics></math> is within a constant factor depending only on <i>d</i> of <math>\n <semantics>\n <msup>\n <mi>N</mi>\n <mfrac>\n <mi>d</mi>\n <mrow>\n <mn>2</mn>\n <mi>d</mi>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n </mfrac>\n </msup>\n <annotation>$N^{\\frac{d}{2d+2}}$</annotation>\n </semantics></math>. This extends the case <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$d=1$</annotation>\n </semantics></math>, which is a celebrated result of Roth and of Matoušek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valkó from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12237","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We prove that the discrepancy of arithmetic progressions in the d-dimensional grid is within a constant factor depending only on d of . This extends the case , which is a celebrated result of Roth and of Matoušek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valkó from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases.
我们证明，在 d 维网格 { 1 , ⋯ , N } d $lbrace 1, \dots, N\rbrace ^d$ 中，算术级数的差异在一个恒定因子之内，这个因子只取决于 N d 2 d + 2 $N^{\frac{d}{2d+2}}$ 的 d。这就扩展了 d = 1 $d=1$ 的情况，这是罗斯以及马图谢克和斯宾塞的著名结果，并且消除了瓦尔科在大约二十年前的上界中的多对数因子。在许多情况下，我们进一步证明了边长不同的网格也有类似的紧约束。
期刊介绍：
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
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