MathematikaPub Date : 2026-03-20DOI: 10.1112/mtk.70083
Noy Soffer Aranov, Angelot Behajaina
{"title":"Counting problems for orthogonal sets and sublattices in function fields","authors":"Noy Soffer Aranov, Angelot Behajaina","doi":"10.1112/mtk.70083","DOIUrl":"https://doi.org/10.1112/mtk.70083","url":null,"abstract":"<p>Let <span></span><math></math>. Analogous to orthogonality in the Euclidean space <span></span><math></math>, there exists a well-studied notion of ultrametric orthogonality in <span></span><math></math>. In this paper, we extend the work of [4] on counting problems related to orthogonality in <span></span><math></math>. For example, we resolve an open question posed in [4] by bounding the size of the largest “orthogonal sets” in <span></span><math></math>. Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over <span></span><math></math>. Finally, we also use ultrametric orthogonality to compute the number of sublattices of <span></span><math></math> with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in <span></span><math></math>. The resulting formulas depend crucially on successive minima.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70083","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147567416","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 : 2026-03-20DOI: 10.1112/mtk.70083
Noy Soffer Aranov, Angelot Behajaina
{"title":"Counting problems for orthogonal sets and sublattices in function fields","authors":"Noy Soffer Aranov, Angelot Behajaina","doi":"10.1112/mtk.70083","DOIUrl":"https://doi.org/10.1112/mtk.70083","url":null,"abstract":"<p>Let <span></span><math></math>. Analogous to orthogonality in the Euclidean space <span></span><math></math>, there exists a well-studied notion of ultrametric orthogonality in <span></span><math></math>. In this paper, we extend the work of [4] on counting problems related to orthogonality in <span></span><math></math>. For example, we resolve an open question posed in [4] by bounding the size of the largest “orthogonal sets” in <span></span><math></math>. Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over <span></span><math></math>. Finally, we also use ultrametric orthogonality to compute the number of sublattices of <span></span><math></math> with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in <span></span><math></math>. The resulting formulas depend crucially on successive minima.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70083","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147567264","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 : 2026-03-19DOI: 10.1112/mtk.70085
Gergely Ambrus, Rainie Heck
{"title":"A note on the Steinitz lemma","authors":"Gergely Ambrus, Rainie Heck","doi":"10.1112/mtk.70085","DOIUrl":"https://doi.org/10.1112/mtk.70085","url":null,"abstract":"<p>We establish the connection between the Steinitz problem for ordering vector families in arbitrary norms and its variant for not necessarily zero-sum families consisting of “nearly unit” vectors.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147567168","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 : 2026-03-19DOI: 10.1112/mtk.70085
Gergely Ambrus, Rainie Heck
{"title":"A note on the Steinitz lemma","authors":"Gergely Ambrus, Rainie Heck","doi":"10.1112/mtk.70085","DOIUrl":"https://doi.org/10.1112/mtk.70085","url":null,"abstract":"<p>We establish the connection between the Steinitz problem for ordering vector families in arbitrary norms and its variant for not necessarily zero-sum families consisting of “nearly unit” vectors.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147567060","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 : 2026-03-06DOI: 10.1112/mtk.70081
Tian Wang, Pengcheng Zhang
{"title":"Ordinary primes for -type abelian varieties and weight 2 modular forms","authors":"Tian Wang, Pengcheng Zhang","doi":"10.1112/mtk.70081","DOIUrl":"https://doi.org/10.1112/mtk.70081","url":null,"abstract":"<p>Let <span></span><math></math> be a <span></span><math></math>-dimensional abelian variety defined over a number field <span></span><math></math>. It is conjectured that the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density, and this is known to be true when <span></span><math></math>, or for certain abelian varieties with extra endomorphisms. In this paper, we extend the family of abelian varieties whose sets of ordinary primes have positive density. Specifically, we show that if the endomorphism algebra of <span></span><math></math> contains a number field <span></span><math></math> of degree <span></span><math></math>, then under certain conditions on the fields <span></span><math></math> and <span></span><math></math>, the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density. This includes <span></span><math></math>-type abelian varieties over <span></span><math></math> (resp., quadratic number fields) of dimension <span></span><math></math> or <span></span><math></math> (resp., <span></span><math></math>) for any rational prime <span></span><math></math>. The proof is carried out in the general setting of compatible systems of Galois representations, and as a consequence, it also implies a positive density result for the sets of ordinary primes of certain modular forms of weight 2.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563972","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 : 2026-03-06DOI: 10.1112/mtk.70081
Tian Wang, Pengcheng Zhang
{"title":"Ordinary primes for -type abelian varieties and weight 2 modular forms","authors":"Tian Wang, Pengcheng Zhang","doi":"10.1112/mtk.70081","DOIUrl":"https://doi.org/10.1112/mtk.70081","url":null,"abstract":"<p>Let <span></span><math></math> be a <span></span><math></math>-dimensional abelian variety defined over a number field <span></span><math></math>. It is conjectured that the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density, and this is known to be true when <span></span><math></math>, or for certain abelian varieties with extra endomorphisms. In this paper, we extend the family of abelian varieties whose sets of ordinary primes have positive density. Specifically, we show that if the endomorphism algebra of <span></span><math></math> contains a number field <span></span><math></math> of degree <span></span><math></math>, then under certain conditions on the fields <span></span><math></math> and <span></span><math></math>, the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density. This includes <span></span><math></math>-type abelian varieties over <span></span><math></math> (resp., quadratic number fields) of dimension <span></span><math></math> or <span></span><math></math> (resp., <span></span><math></math>) for any rational prime <span></span><math></math>. The proof is carried out in the general setting of compatible systems of Galois representations, and as a consequence, it also implies a positive density result for the sets of ordinary primes of certain modular forms of weight 2.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563973","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 : 2026-03-05DOI: 10.1112/mtk.70080
Dung M. Ha, Anh D. Hoang, Hieu T. Ngo
{"title":"On the least almost-prime in an arithmetic progression","authors":"Dung M. Ha, Anh D. Hoang, Hieu T. Ngo","doi":"10.1112/mtk.70080","DOIUrl":"https://doi.org/10.1112/mtk.70080","url":null,"abstract":"<p>For all sufficiently large <span></span><math></math>, in any arithmetic progression <span></span><math></math> in which <span></span><math></math> and <span></span><math></math> are relatively prime there exists a positive integer with at most two prime factors (counted with multiplicity) which is asymptotically less than <span></span><math></math>. The proof uses the weighted sieve of Greaves–Halberstam–Richert with bilinear remainder terms and Selberg's sieve.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563490","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 : 2026-03-05DOI: 10.1112/mtk.70080
Dung M. Ha, Anh D. Hoang, Hieu T. Ngo
{"title":"On the least almost-prime in an arithmetic progression","authors":"Dung M. Ha, Anh D. Hoang, Hieu T. Ngo","doi":"10.1112/mtk.70080","DOIUrl":"https://doi.org/10.1112/mtk.70080","url":null,"abstract":"<p>For all sufficiently large <span></span><math></math>, in any arithmetic progression <span></span><math></math> in which <span></span><math></math> and <span></span><math></math> are relatively prime there exists a positive integer with at most two prime factors (counted with multiplicity) which is asymptotically less than <span></span><math></math>. The proof uses the weighted sieve of Greaves–Halberstam–Richert with bilinear remainder terms and Selberg's sieve.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563292","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 : 2026-03-03DOI: 10.1112/mtk.70082
Georgi Vodev
{"title":"Semiclassical resolvent estimates for the magnetic Schrödinger operator","authors":"Georgi Vodev","doi":"10.1112/mtk.70082","DOIUrl":"https://doi.org/10.1112/mtk.70082","url":null,"abstract":"<p>We obtain semiclassical resolvent estimates for the Schrödinger operator <span></span><math></math> in <span></span><math></math>, <span></span><math></math>, where <span></span><math></math> is a semiclassical parameter, <span></span><math></math> and <span></span><math></math> are real-valued electric and magnetic potentials independent of <span></span><math></math>. If <span></span><math></math>, <span></span><math></math>, <span></span><math></math> satisfy <span></span><math></math>, <span></span><math></math>, <span></span><math></math>, <span></span><math></math>, for <span></span><math></math>, we prove that the norm of the weighted resolvent is bounded by <span></span><math></math>, <span></span><math></math>. We get better resolvent bounds for electric potentials which are Hölder with respect to the radial variable and magnetic potentials which are Hölder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form <span></span><math></math>, <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563020","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 : 2026-03-03DOI: 10.1112/mtk.70082
Georgi Vodev
{"title":"Semiclassical resolvent estimates for the magnetic Schrödinger operator","authors":"Georgi Vodev","doi":"10.1112/mtk.70082","DOIUrl":"https://doi.org/10.1112/mtk.70082","url":null,"abstract":"<p>We obtain semiclassical resolvent estimates for the Schrödinger operator <span></span><math></math> in <span></span><math></math>, <span></span><math></math>, where <span></span><math></math> is a semiclassical parameter, <span></span><math></math> and <span></span><math></math> are real-valued electric and magnetic potentials independent of <span></span><math></math>. If <span></span><math></math>, <span></span><math></math>, <span></span><math></math> satisfy <span></span><math></math>, <span></span><math></math>, <span></span><math></math>, <span></span><math></math>, for <span></span><math></math>, we prove that the norm of the weighted resolvent is bounded by <span></span><math></math>, <span></span><math></math>. We get better resolvent bounds for electric potentials which are Hölder with respect to the radial variable and magnetic potentials which are Hölder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form <span></span><math></math>, <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563019","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}