{"title":"Two-dimensional Weyl sums failing square-root cancellation along lines","authors":"Julia Brandes, Igor E. Shparlinski","doi":"10.4310/arkiv.2023.v61.n2.a1","DOIUrl":null,"url":null,"abstract":"We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arkiv for Matematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/arkiv.2023.v61.n2.a1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.
我们证明了长度为$P$的二维Weyl和族在几乎所有单位环面线性片上的值都高达$P^{3/4 + o(1)}$,这与人们普遍认为Weyl和在单位环面的一般子变异上应该表现平方根消去的期望相矛盾。这是J. Brandes, S. T. Parsell, C. Poulias, G. Shakan和R. C. Vaughan(2020)从二次多项式和三次多项式到任意次一般多项式的结果的推广。我们方法的新成分是E. Bombieri(1966)关于曲线上的指数和的经典结果,以及R. J. Duffin和a . C. Schaeffer(1941)关于以素数为分母的有理数的Diophantine近似的经典结果。