Estimates for smooth Weyl sums on minor arcs

IF 0.9 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-12-19 DOI:10.1112/blms.13219

Jörg Brüdern, Trevor D. Wooley

{"title":"Estimates for smooth Weyl sums on minor arcs","authors":"Jörg Brüdern, Trevor D. Wooley","doi":"10.1112/blms.13219","DOIUrl":null,"url":null,"abstract":"We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <msup>\n <mi>n</mi>\n <mi>k</mi>\n </msup>\n </mrow>\n <annotation>$\\alpha n^k$</annotation>\n </semantics></math>. In particular, when <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n <mo>⩾</mo>\n <mn>6</mn>\n </mrow>\n <annotation>$k\\geqslant 6$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>ρ</mi>\n <mo>(</mo>\n <mi>k</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\rho (k)$</annotation>\n </semantics></math> is defined via the relation <math>\n <semantics>\n <mrow>\n <mi>ρ</mi>\n <msup>\n <mrow>\n <mo>(</mo>\n <mi>k</mi>\n <mo>)</mo>\n </mrow>\n <mrow>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>=</mo>\n <mi>k</mi>\n <mrow>\n <mo>(</mo>\n <mi>log</mi>\n <mi>k</mi>\n <mo>+</mo>\n <mn>8.02113</mn>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\rho (k)^{-1}=k(\\log k+8.02113)$</annotation>\n </semantics></math>, then for all large numbers <math>\n <semantics>\n <mi>N</mi>\n <annotation>$N$</annotation>\n </semantics></math> there is an integer <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>⩽</mo>\n <mi>n</mi>\n <mo>⩽</mo>\n <mi>N</mi>\n </mrow>\n <annotation>$1\\leqslant n\\leqslant N$</annotation>\n </semantics></math> for which <math>\n <semantics>\n <mrow>\n <mrow>\n <mo>∥</mo>\n <mi>α</mi>\n </mrow>\n <msup>\n <mi>n</mi>\n <mi>k</mi>\n </msup>\n <mrow>\n <mo>∥</mo>\n <mo>⩽</mo>\n </mrow>\n <msup>\n <mi>N</mi>\n <mrow>\n <mo>−</mo>\n <mi>ρ</mi>\n <mo>(</mo>\n <mi>k</mi>\n <mo>)</mo>\n </mrow>\n </msup>\n </mrow>\n <annotation>$\\Vert \\alpha n^k\\Vert \\leqslant N^{-\\rho (k)}$</annotation>\n </semantics></math>.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"657-668"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13219","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.13219","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of $α n^{k}$ . In particular, when $k ⩾ 6$ and $ρ (k)$ is defined via the relation $ρ {(k)}^{- 1} = k (\log k + 8.02113)$ , then for all large numbers $N$ there is an integer $n$ with $1 ⩽ n ⩽ N$ for which $∥ α n^{k} ∥ ⩽ N^{- ρ (k)}$ .

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

小弧上光滑Weyl和的估计

我们提供了小弧上光滑Weyl和的新估计，并探讨了它们对α n k的分数部分分布的影响$\alpha n^k$。特别是，当k小于6 $k\geqslant 6$和ρ (k) $\rho (k)$通过关系ρ定义时(k)−1 = k （log k + 8.02113）$\rho (k)^{-1}=k(\log k+8.02113)$，那么对于所有的大数N $N$，有一个整数N $n$, 1≤N≤N $1\leqslant n\leqslant N$∥α n k∥n−ρ (k)$\Vert \alpha n^k\Vert \leqslant N^{-\rho (k)}$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊