{"title":"绕过高斯圆问题：哈代猜想与圆附近网格点的分布","authors":"Stephen Lester, Igor Wigman","doi":"10.1112/jlms.12977","DOIUrl":null,"url":null,"abstract":"<p>Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-<span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> disc by its area is <span></span><math>\n <semantics>\n <mrow>\n <mi>O</mi>\n <mo>(</mo>\n <msup>\n <mi>R</mi>\n <mrow>\n <mn>1</mn>\n <mo>/</mo>\n <mn>2</mn>\n <mo>+</mo>\n <mi>o</mi>\n <mo>(</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n </msup>\n <mo>)</mo>\n </mrow>\n <annotation>$O(R^{1/2+o(1)})$</annotation>\n </semantics></math>. One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 3","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12977","citationCount":"0","resultStr":"{\"title\":\"Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles\",\"authors\":\"Stephen Lester, Igor Wigman\",\"doi\":\"10.1112/jlms.12977\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-<span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> disc by its area is <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>O</mi>\\n <mo>(</mo>\\n <msup>\\n <mi>R</mi>\\n <mrow>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mn>2</mn>\\n <mo>+</mo>\\n <mi>o</mi>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$O(R^{1/2+o(1)})$</annotation>\\n </semantics></math>. One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"110 3\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12977\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12977\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12977","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
哈代猜想，用一个半径为 R $R$ 的圆盘的面积来近似位于该圆盘中的晶格点的数量，其误差项为 O ( R 1 / 2 + o ( 1 ) ) $O(R^{1/2+o(1)})$。支持这一猜想的一个来源是一种民间启发式，它使用 i.i.d. 随机变量来模拟位于边界附近的晶格点，并对这些随机变量的和进行平方根抵消。我们通过研究这些网格点如何相互影响来检验这一启发式，并证明它们的自相关性是由随机模型决定的。此外，我们还证明了边界附近 "分离得很好 "的网格点的独立行为。我们还提出了关于这些晶格点的成对分布的猜想。
Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles
Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius- disc by its area is . One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.
期刊介绍：
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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