MathematikaPub Date : 2026-04-08DOI: 10.1112/mtk.70092
Spencer Bullent
{"title":"The Steklov spectrum of spherical cylinders","authors":"Spencer Bullent","doi":"10.1112/mtk.70092","DOIUrl":"10.1112/mtk.70092","url":null,"abstract":"<p>The Steklov problem on a compact Lipschitz domain is to find harmonic functions on the interior whose outward normal derivative on the boundary is some multiple (eigenvalue) of their trace on the boundary. These eigenvalues form the Steklov spectrum of the domain. This article considers the Steklov spectrum of spherical cylinders (Euclidean ball times interval). It is shown that the spectral counting function admits a two-term asymptotic expansion. The coefficient of the second term consists of a contribution from the curvature of the boundary and a contribution from the edges.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70092","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683511","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}
MathematikaPub Date : 2026-04-06DOI: 10.1112/mtk.70089
Vasiliy Neckrasov
{"title":"Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle","authors":"Vasiliy Neckrasov","doi":"10.1112/mtk.70089","DOIUrl":"10.1112/mtk.70089","url":null,"abstract":"<p>In 2019 Kleinbock and Wadleigh proved a “zero-one law” for uniform inhomogeneous Diophantine approximations. We generalize this statement to arbitrary weight functions and establish a new and simple proof of this statement, based on the transference principle. We also give a complete description of the sets of <span></span><math></math>-Dirichlet pairs with a fixed matrix in this setthe up from Lebesgue measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work (and sometimes can be strengthened) for weighted Diophantine approximations.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70089","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683299","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}
MathematikaPub Date : 2026-04-03DOI: 10.1112/mtk.70084
Gaurav Aggarwal
{"title":"Dimension bounds for singular affine forms","authors":"Gaurav Aggarwal","doi":"10.1112/mtk.70084","DOIUrl":"10.1112/mtk.70084","url":null,"abstract":"<p>In this paper, we establish upper bounds on the dimension of sets of singular-on-average and <span></span><math></math>-singular affine forms in singly metric settings where either the matrix or the shift is fixed. These results partially address open questions posed by Das, Fishman, Simmons, and Urbański, as well as Kleinbock and Wadleigh. Furthermore, we extend our results to the generalized weighted setup and derive bounds for the intersection of these sets with a wide class of fractals.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683130","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}
MathematikaPub Date : 2026-04-01DOI: 10.1112/mtk.70090
Jaehyeon Ryu, Andreas Seeger
{"title":"Failure of stability of a maximal operator bound for perturbed Nevo–Thangavelu means","authors":"Jaehyeon Ryu, Andreas Seeger","doi":"10.1112/mtk.70090","DOIUrl":"10.1112/mtk.70090","url":null,"abstract":"<p>Let <span></span><math></math> be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra <span></span><math></math>, where <span></span><math></math>. We consider maximal functions associated to spheres in a <span></span><math></math>-dimensional linear subspace <span></span><math></math>, dilated by the automorphic dilations. <span></span><math></math> boundedness results for the case where <span></span><math></math> are well understood. Here, we consider the case of a tilted hyperplane <span></span><math></math> which is not invariant under the automorphic dilations. In the case of Métivier groups, it is known that the <span></span><math></math>-boundedness results are stable under a small linear tilt. We show that this is generally not the case for other two-step groups, and provide new necessary conditions for <span></span><math></math> boundedness. We prove these results in a more general setting with tilted versions of submanifolds of <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70090","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147682875","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}
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