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On absolutely friendly measures on 绝对友好的措施
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-24 DOI: 10.1112/mtk.12256
Shreyasi Datta, Justin Liu
{"title":"On absolutely friendly measures on","authors":"Shreyasi Datta,&nbsp;Justin Liu","doi":"10.1112/mtk.12256","DOIUrl":"https://doi.org/10.1112/mtk.12256","url":null,"abstract":"<p>In this paper, we extend the work of Pollington and Velani [Selecta Math. 11(2005)] to an <span></span><math></math>-arithmetic set-up, where <span></span><math></math> is a finite set of valuations of <span></span><math></math>. In particular, for an <i>absolutely friendly</i> measure <span></span><math></math> supported on a compact set in <span></span><math></math>, we give a summation condition on an approximating function <span></span><math></math> such that <span></span><math></math> almost no point in the compact set is <span></span><math></math> approximable. The crucial ingredient is a version of the simplex lemma that we prove dynamically.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12256","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Product-free sets in the free group 自由组中的无积集合
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-18 DOI: 10.1112/mtk.12255
Miquel Ortega, Juanjo Rué, Oriol Serra
{"title":"Product-free sets in the free group","authors":"Miquel Ortega,&nbsp;Juanjo Rué,&nbsp;Oriol Serra","doi":"10.1112/mtk.12255","DOIUrl":"https://doi.org/10.1112/mtk.12255","url":null,"abstract":"<p>We prove that product-free subsets of the free group over a finite alphabet have maximum upper density <span></span><math></math> with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly <span></span><math></math>-product-free sets have maximum upper density <span></span><math></math> in terms of this measure. The bounds are tight.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141069124","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}
引用次数: 0
Triangular maximal operators on locally finite trees 局部有限树上的三角最大算子
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-15 DOI: 10.1112/mtk.12253
Stefano Meda, Federico Santagati
{"title":"Triangular maximal operators on locally finite trees","authors":"Stefano Meda,&nbsp;Federico Santagati","doi":"10.1112/mtk.12253","DOIUrl":"https://doi.org/10.1112/mtk.12253","url":null,"abstract":"<p>We introduce the centred and the uncentred triangular maximal operators <span></span><math></math> and <span></span><math></math>, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both <span></span><math></math> and <span></span><math></math> are bounded on <span></span><math></math> for every <span></span><math></math> in <span></span><math></math>, that <span></span><math></math> is also bounded on <span></span><math></math>, and that <span></span><math></math> is not of weak type (1, 1) on homogeneous trees. Our proof of the <span></span><math></math> boundedness of <span></span><math></math> hinges on the geometric approach of Córdoba and Fefferman. We also establish <span></span><math></math> bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy–Littlewood maximal operators (on balls) may be unbounded on <span></span><math></math> for every <span></span><math></math> even on some trees where the number of neighbours is uniformly bounded.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12253","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140924815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a lower bound of Hausdorff dimension of weighted singular vectors 论加权奇异向量的 Hausdorff 维数下限
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-14 DOI: 10.1112/mtk.12252
Taehyeong Kim, Jaemin Park
{"title":"On a lower bound of Hausdorff dimension of weighted singular vectors","authors":"Taehyeong Kim,&nbsp;Jaemin Park","doi":"10.1112/mtk.12252","DOIUrl":"https://doi.org/10.1112/mtk.12252","url":null,"abstract":"<p>Let <span></span><math></math> be a <span></span><math></math>-tuple of positive real numbers such that <span></span><math></math> and <span></span><math></math>. A <span></span><math></math>-dimensional vector <span></span><math></math> is said to be <span></span><math></math>-singular if for every <span></span><math></math>, there exists <span></span><math></math> such that for all <span></span><math></math>, the system of inequalities\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140924830","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}
引用次数: 0
Sharpening Littlewood subordination principle with univalent symbol 用单价符号强化利特尔伍德从属性原理
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-11 DOI: 10.1112/mtk.12254
Dušica Dmitrović, Boban Karapetrović
{"title":"Sharpening Littlewood subordination principle with univalent symbol","authors":"Dušica Dmitrović,&nbsp;Boban Karapetrović","doi":"10.1112/mtk.12254","DOIUrl":"https://doi.org/10.1112/mtk.12254","url":null,"abstract":"<p>Let <span></span><math></math> be a subharmonic function on the open unit disc <span></span><math></math>, centered at the origin of the complex plane, and let <span></span><math></math> be a holomorphic function such that <span></span><math></math>. A classical result, known as Littlewood subordination principle, states <span></span><math></math>, where <span></span><math></math> and <span></span><math></math> are integral means over the circle of radius <span></span><math></math> centered at the origin, of the functions <span></span><math></math> and <span></span><math></math>, respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function <span></span><math></math> is univalent, by proving that\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140910577","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}
引用次数: 0
Large deviations of the argument of the Riemann zeta function 黎曼zeta函数参数的大偏差
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-07 DOI: 10.1112/mtk.12251
Alexander Dobner
{"title":"Large deviations of the argument of the Riemann zeta function","authors":"Alexander Dobner","doi":"10.1112/mtk.12251","DOIUrl":"https://doi.org/10.1112/mtk.12251","url":null,"abstract":"<p>Let <span></span><math></math>. We prove an unconditional lower bound on the measure of the sets <span></span><math></math> for <span></span><math></math>. For <span></span><math></math>, our bound has a Gaussian shape with variance proportional to <span></span><math></math>. At the endpoint, <span></span><math></math>, our result implies the best known <span></span><math></math>-theorem for <span></span><math></math> that is due to Tsang. We also explain how the method breaks down for <span></span><math></math> given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12251","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140844649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Markoff–Lagrange spectrum of one-sided shifts 单边移动的马尔科夫-拉格朗日谱
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-07 DOI: 10.1112/mtk.12250
Hajime Kaneko, Wolfgang Steiner
{"title":"Markoff–Lagrange spectrum of one-sided shifts","authors":"Hajime Kaneko,&nbsp;Wolfgang Steiner","doi":"10.1112/mtk.12250","DOIUrl":"https://doi.org/10.1112/mtk.12250","url":null,"abstract":"<p>For the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are <span></span><math></math>-adic sequences, where <span></span><math></math> is a certain infinite set of substitutions that contains Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff–Lagrange spectrum.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12250","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140844614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mixed-norm estimates via the helicoidal method 通过螺旋线方法进行混合正态估计
IF 0.8 3区 数学
Mathematika Pub Date : 2024-04-21 DOI: 10.1112/mtk.12248
Cristina Benea, Camil Muscalu
{"title":"Mixed-norm estimates via the helicoidal method","authors":"Cristina Benea,&nbsp;Camil Muscalu","doi":"10.1112/mtk.12248","DOIUrl":"https://doi.org/10.1112/mtk.12248","url":null,"abstract":"<p>We prove multiple vector-valued and mixed-norm estimates for multilinear operators in <span></span><math></math>, more precisely for multilinear operators <span></span><math></math> associated to a symbol singular along a <span></span><math></math>-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension <span></span><math></math>, the input functions are not necessarily in <span></span><math></math> and can instead be elements of mixed-norm spaces <span></span><math></math>.</p><p>Such a result has interesting consequences especially when <span></span><math></math> spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy “purely mixed-norm estimates” and no classical <span></span><math></math> estimates.</p><p>Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12248","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140622646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a conjecture of Graham on the -divisibility of central binomial coefficients 格雷厄姆关于中心二项式系数可分性的猜想
IF 0.8 3区 数学
Mathematika Pub Date : 2024-04-18 DOI: 10.1112/mtk.12249
Ernie Croot, Hamed Mousavi, Maxie Schmidt
{"title":"On a conjecture of Graham on the -divisibility of central binomial coefficients","authors":"Ernie Croot,&nbsp;Hamed Mousavi,&nbsp;Maxie Schmidt","doi":"10.1112/mtk.12249","DOIUrl":"https://doi.org/10.1112/mtk.12249","url":null,"abstract":"<p>Let <span></span><math></math> be pairwise distinct primes. From a theorem of Kummer, each prime <span></span><math></math> can divide <span></span><math></math> at most <span></span><math></math> times. We show that, for all <span></span><math></math>, if <span></span><math></math> are sufficiently large in terms of <span></span><math></math> and <span></span><math></math>, then there exist infinitely many positive integers <span></span><math></math> such that each <span></span><math></math> divides <span></span><math></math> at most <span></span><math></math> times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers <span></span><math></math> such that <span></span><math></math> is coprime to 105.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12249","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140619726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Shifted convolution sums for averaged over weighted sets 加权平均集的移位卷积和
IF 0.8 3区 数学
Mathematika Pub Date : 2024-03-29 DOI: 10.1112/mtk.12247
Wing Hong Leung
{"title":"Shifted convolution sums for averaged over weighted sets","authors":"Wing Hong Leung","doi":"10.1112/mtk.12247","DOIUrl":"https://doi.org/10.1112/mtk.12247","url":null,"abstract":"<p>Let <span></span><math></math> be the Fourier coefficients of an <span></span><math></math> Hecke–Maass cusp form <span></span><math></math> and <span></span><math></math> be those of an <span></span><math></math> Hecke holomorphic or Hecke–Maass cusp form <span></span><math></math>. Let <span></span><math></math> and <span></span><math></math> be a sequence. We show that if <span></span><math></math> for some <span></span><math></math>,\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12247","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140321771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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