Mathematika最新文献

{"title":"The theorem modulo a prime: High density for","authors":"David J. Grynkiewicz","doi":"10.1112/mtk.70030","DOIUrl":"https://doi.org/10.1112/mtk.70030","url":null,"abstract":"<p>The <span></span><math></math> Theorem for <span></span><math></math> asserts that, if <span></span><math></math> are finite, nonempty subsets with <span></span><math></math> and <span></span><math></math>, then there exist arithmetic progressions <span></span><math></math> and <span></span><math></math> of common difference such that <span></span><math></math> and <span></span><math></math> for all <span></span><math></math>. These are instances of Freiman's theorem with precise bounds. There is much partial progress extending this result to nonempty subsets <span></span><math></math> with <span></span><math></math> prime, <span></span><math></math> and <span></span><math></math>. The ideal conjectured density restriction under which such a version of the <span></span><math></math> Theorem modulo <span></span><math></math> is expected is <span></span><math></math>. Under this ideal density constraint, we show that there are arithmetic progressions <span></span><math></math>, <span></span><math></math>, and <span></span><math></math> of common difference with <span></span><math></math> and <span></span><math></math> for all <span></span><math></math>, where <span></span><math></math>, provided <span></span><math></math>. This generalizes a result of Serra and Zémor [33] by extending their work from the special case <span></span><math></math> to that of general sumsets <span></span><math></math>, removes all unnecessary sufficiently large <span></span><math></math> restrictions, and improves (even in the case <span></span><math></math>) their constant 100-fold, from 0.0001 to 0.01. As part of the proof, we additionally obtain a yet better 1000-fold improvement of their constants at the cost of a near optimal density restriction of the form <span></span><math></math> (Theorem 3.5 and Corollary 3.7). These give rare high-density versions of the <span></span><math></math> Theorem for general sumsets <span></span><math></math> modulo <span></span><math></math> and are the first instances with tangible (rather than effectively existential) values for the constants for general sumsets <span></span><math></math> with high density, or indeed for any density without added constraints on the relative sizes of <span></span><math></math> and <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144308962","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":"Arithmetic constants for symplectic variances of the divisor function","authors":"Vivian Kuperberg,&nbsp;Matilde Lalín","doi":"10.1112/mtk.70029","DOIUrl":"https://doi.org/10.1112/mtk.70029","url":null,"abstract":"<p>Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function <span></span><math></math> over number fields, which were inspired by analogous results over function fields proven by the authors. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70029","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144273512","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}

{"title":"Distribution of local signs of modular forms and murmurations of Fourier coefficients","authors":"Kimball Martin","doi":"10.1112/mtk.70028","DOIUrl":"https://doi.org/10.1112/mtk.70028","url":null,"abstract":"<p>Recently, we showed that global root numbers of modular forms are biased toward <span></span><math></math>. Together with Pharis, we also showed an initial bias of Fourier coefficients toward the sign of the root number. First, we prove analogous results with respect to local root numbers. Second, a subtle correlation between Fourier coefficients and global root numbers, termed murmurations, was recently discovered for elliptic curves and modular forms. We conjecture murmurations in a more general context of different (possibly empty) combinations of local root numbers. Last, the Appendix corrects a sign error in our joint paper with Pharis.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144197525","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":"Maximal -subsets of manifolds","authors":"Ciprian Demeter,&nbsp;Hongki Jung,&nbsp;Donggeun Ryou","doi":"10.1112/mtk.70026","DOIUrl":"https://doi.org/10.1112/mtk.70026","url":null,"abstract":"<p>We construct maximal <span></span><math></math>-subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents <span></span><math></math>. Our arguments combine restriction estimates and decoupling with old and new probabilistic estimates.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144148647","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":"Mills' constant is irrational","authors":"Kota Saito","doi":"10.1112/mtk.70027","DOIUrl":"https://doi.org/10.1112/mtk.70027","url":null,"abstract":"<p>Let <span></span><math></math> denote the integer part of <span></span><math></math>. In 1947, Mills constructed a real number <span></span><math></math> such that <span></span><math></math> is always a prime number for every positive integer <span></span><math></math>. We define Mills' constant as the smallest real number <span></span><math></math> satisfying this property. Determining whether this number is irrational has been a long-standing problem. In this paper, we show that Mills' constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144148382","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":"Non-autonomous iteration of polynomials in the complex plane","authors":"Marta Kosek,&nbsp;Małgorzata Stawiska","doi":"10.1112/mtk.70025","DOIUrl":"https://doi.org/10.1112/mtk.70025","url":null,"abstract":"<p>We consider a sequence <span></span><math></math> of polynomials with uniformly bounded zeros and <span></span><math></math>, <span></span><math></math> for <span></span><math></math>, satisfying certain asymptotic conditions. We prove that the function sequence <span></span><math></math> is uniformly convergent in <span></span><math></math>. The non-autonomous filled Julia set <span></span><math></math> generated by the polynomial sequence <span></span><math></math> is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by <span></span><math></math>, where <span></span><math></math> is the classical Chebyshev polynomial of degree <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144125999","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 orthogonal and staircase connectedness in the plane","authors":"Julia Q. Du,&nbsp;Liping Yuan,&nbsp;Tudor Zamfirescu","doi":"10.1112/mtk.70021","DOIUrl":"https://doi.org/10.1112/mtk.70021","url":null,"abstract":"<p>In this paper, we introduce <i>o</i>-extreme points defined by using orthogonal paths in orthogonally connected sets. We investigate their properties and obtain Minkowski-type theorems involving orthogonally connected sets. Using <i>o</i>-extreme points, we give some characterizations of staircase connectedness.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143888809","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 note on limsup sets of annuli","authors":"Mumtaz Hussain,&nbsp;Benjamin Ward","doi":"10.1112/mtk.70023","DOIUrl":"https://doi.org/10.1112/mtk.70023","url":null,"abstract":"<p>We consider the set of points in infinitely many max-norm annuli centred at rational points in <span></span><math></math>. We give Jarník–Besicovitch-type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension <span></span><math></math>, and the thickness of the annuli is decreasing rapidly, then the dimension of the set tends towards <span></span><math></math>. We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's scaling lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021).</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143880056","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":"Minimal periodic foams with fixed inradius","authors":"Annalisa Cesaroni,&nbsp;Matteo Novaga","doi":"10.1112/mtk.70020","DOIUrl":"https://doi.org/10.1112/mtk.70020","url":null,"abstract":"<p>In this note, we show existence and regularity of periodic tilings of the Euclidean space into equal cells containing a ball of fixed radius, which minimize either the classical or the fractional perimeter. We also discuss some qualitative properties of minimizers in dimensions 3 and 4.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143861591","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}

{"title":"Spherical functions and Stolarsky's invariance principle","authors":"M. M. Skriganov","doi":"10.1112/mtk.70019","DOIUrl":"https://doi.org/10.1112/mtk.70019","url":null,"abstract":"<p>In the previous paper (Skriganov, <i>J. Complexity</i> 56 (2020), 101428), Stolarsky's invariance principle, known in the literature for point distributions on Euclidean spheres, has been extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. Geometric features of these spaces as well as their models in terms of Jordan algebras have been used very essentially in the proof. In the present paper, a new pure analytic proof of the extended Stolarsky's invariance principle is given, relying on the theory of spherical functions on compact Riemannian symmetric manifolds of rank one.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143853058","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}