{"title":"Hybrid Bounds on Two-Parametric Families of Weyl Sums Along Smooth Curves","authors":"Chang-Pao Chen, I. Shparlinski","doi":"10.1307/mmj/20205904","DOIUrl":"https://doi.org/10.1307/mmj/20205904","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76850854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ga3-Structures on del Pezzo Fibrations","authors":"Masaru Nagaoka","doi":"10.1307/MMJ/20195835","DOIUrl":"https://doi.org/10.1307/MMJ/20195835","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"10 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85898395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let p be a prime number. We give several results towards a particular instance of a conjecture of Einsiedler and Kleinbock asserting that every p-adic number x satisfies inf a,b∈Z∖{0}|ab|⋅|ax−b|p=0. We highlight a close relationship between this conjecture and the (still open) p-adic Littlewood conjecture, according to which every real number ξ satisfies inf q∈Z,q⩾1q⋅‖qξ‖⋅|q|p=0. Furthermore, we discuss the analogues of these conjectures over fields of power series.
{"title":"Multiplicative p -Adic Approximation","authors":"D. Badziahin, Y. Bugeaud","doi":"10.1307/mmj/20195785","DOIUrl":"https://doi.org/10.1307/mmj/20195785","url":null,"abstract":"Let p be a prime number. We give several results towards a particular instance of a conjecture of Einsiedler and Kleinbock asserting that every p-adic number x satisfies inf a,b∈Z∖{0}|ab|⋅|ax−b|p=0. We highlight a close relationship between this conjecture and the (still open) p-adic Littlewood conjecture, according to which every real number ξ satisfies inf q∈Z,q⩾1q⋅‖qξ‖⋅|q|p=0. Furthermore, we discuss the analogues of these conjectures over fields of power series.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"8 1","pages":"1-23"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86613645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study a Hamilton–Jacobi flow {Htτ}t>0 starting from the Takagi function τ. The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on R. As the first result of this paper, we derive an explicit representation of {Htτ}. It turns out that Htτ is a piecewise quadratic function at any time and that the points of intersection between the parabolas are given in terms of binary expansion of real numbers. Applying the representation formula, we next give the main result, which asserts that {Htτ} has a self-affine property of evolutional type involving a time difference in the functional equality. Furthermore, we determine the optimal time until when the self-affine property is valid.
{"title":"A Self-Affine Property of Evolutional Type Appearing in a Hamilton–Jacobi Flow Starting from the Takagi Function","authors":"Y. Fujita, N. Hamamuki, Norikazu Yamaguchi","doi":"10.1307/mmj/20195782","DOIUrl":"https://doi.org/10.1307/mmj/20195782","url":null,"abstract":"In this paper, we study a Hamilton–Jacobi flow {Htτ}t>0 starting from the Takagi function τ. The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on R. As the first result of this paper, we derive an explicit representation of {Htτ}. It turns out that Htτ is a piecewise quadratic function at any time and that the points of intersection between the parabolas are given in terms of binary expansion of real numbers. Applying the representation formula, we next give the main result, which asserts that {Htτ} has a self-affine property of evolutional type involving a time difference in the functional equality. Furthermore, we determine the optimal time until when the self-affine property is valid.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"11 1","pages":"1-16"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91157733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Fixed Point Method and Bloch’s Theorem","authors":"J. Cortissoz","doi":"10.1307/MMJ/20195829","DOIUrl":"https://doi.org/10.1307/MMJ/20195829","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"12 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79550271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Lower Bound for the Hausdorff Dimension of the Green Current","authors":"H. Thelin","doi":"10.1307/MMJ/20195844","DOIUrl":"https://doi.org/10.1307/MMJ/20195844","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"21 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78039020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Fine Property of Sets at Points of Lebesgue Density","authors":"S. Delladio","doi":"10.1307/mmj/20205874","DOIUrl":"https://doi.org/10.1307/mmj/20205874","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"74 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72807455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Effect of Cell-Attachment on the Group of Self-Equivalences of an Elliptic Space","authors":"M. Benkhalifa","doi":"10.1307/MMJ/20195840","DOIUrl":"https://doi.org/10.1307/MMJ/20195840","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"11 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79518228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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