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Bounded diameter monochromatic component covers 有界直径单色元件盖
IF 0.6 3区 数学
Mathematika Pub Date : 2026-08-20 DOI: 10.1112/mtk.70123
Alexey Pokrovskiy
{"title":"Bounded diameter monochromatic component covers","authors":"Alexey Pokrovskiy","doi":"10.1112/mtk.70123","DOIUrl":"https://doi.org/10.1112/mtk.70123","url":null,"abstract":"<p>Ryser conjectured that every <span></span><math></math>-edge-coloured complete graph can be covered by <span></span><math></math> monochromatic trees. Motivated by a question of Austin in analysis, Milićević predicted something stronger — that every <span></span><math></math>-edge-coloured complete graph can be covered by <span></span><math></math> monochromatic trees <i>of bounded diameter</i>. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Milićević's Conjecture, most notably that it is true for <span></span><math></math>. We also obtain several new cases of a generalization of Milićević's Conjecture to non-complete graphs due to DeBiasio–Kamel–McCourt–Sheats.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70123","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148784907","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
Buffon discrepancy and the Steinhaus longimeter 布冯的差异和施泰因豪斯的长度
IF 0.6 3区 数学
Mathematika Pub Date : 2026-08-13 DOI: 10.1112/mtk.70126
Stefan Steinerberger
{"title":"Buffon discrepancy and the Steinhaus longimeter","authors":"Stefan Steinerberger","doi":"10.1112/mtk.70126","DOIUrl":"https://doi.org/10.1112/mtk.70126","url":null,"abstract":"<p>Let <span></span><math></math> be a convex set. We study the problem of distributing a one-dimensional set <span></span><math></math> with total length <span></span><math></math> so that for any line <span></span><math></math> in <span></span><math></math> the number of intersections <span></span><math></math> is proportional to the length <span></span><math></math> as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy <span></span><math></math>. We also show that the unit disk <span></span><math></math> admits a set with uniformly bounded Buffon discrepancy as <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148754084","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
Zeros of polynomials in derivatives of automorphic -functions 自同构函数的导数中多项式的零
IF 0.6 3区 数学
Mathematika Pub Date : 2026-07-30 DOI: 10.1112/mtk.70120
Anji Dong, Nawapan Wattanawanichkul, Alexandru Zaharescu
{"title":"Zeros of polynomials in derivatives of automorphic -functions","authors":"Anji Dong,&nbsp;Nawapan Wattanawanichkul,&nbsp;Alexandru Zaharescu","doi":"10.1112/mtk.70120","DOIUrl":"https://doi.org/10.1112/mtk.70120","url":null,"abstract":"<p>Let <span></span><math></math> be the set of all cuspidal automorphic representations of <span></span><math></math>, and let <span></span><math></math> be a polynomial in the derivatives of <span></span><math></math>-functions associated with representations <span></span><math></math>. We establish an asymptotic formula for the number of nontrivial zeros of <span></span><math></math> with <span></span><math></math>. We explicitly determine the main term of this formula in terms of the dimensions, the arithmetic conductors, and the orders of differentiation of the component <span></span><math></math>-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of <span></span><math></math> lie near the critical line <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70120","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148617049","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
Abelian number fields with frobenian conditions 具有frobenian条件的Abelian数域
IF 0.6 3区 数学
Mathematika Pub Date : 2026-07-28 DOI: 10.1112/mtk.70119
Julie Tavernier
{"title":"Abelian number fields with frobenian conditions","authors":"Julie Tavernier","doi":"10.1112/mtk.70119","DOIUrl":"https://doi.org/10.1112/mtk.70119","url":null,"abstract":"<p>We study the distribution of abelian number fields with frobenian conditions imposed on the conductor. In particular, we find an asymptotic for the number of abelian field extensions of a number field <span></span><math></math> whose conductor is the sum of two squares. We also discuss an application of the Brauer group of stacks to quadratic number fields.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70119","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616220","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
Bounded exponential sums with multiplicative coefficients 带乘积系数的有界指数和
IF 0.6 3区 数学
Mathematika Pub Date : 2026-07-24 DOI: 10.1112/mtk.70117
Péa Bazin, Ihor Pylaiev, Fred Tyrrell
{"title":"Bounded exponential sums with multiplicative coefficients","authors":"Péa Bazin,&nbsp;Ihor Pylaiev,&nbsp;Fred Tyrrell","doi":"10.1112/mtk.70117","DOIUrl":"https://doi.org/10.1112/mtk.70117","url":null,"abstract":"<p>We investigate when the exponential sum <span></span><math></math> is bounded, for a multiplicative function <span></span><math></math> and <span></span><math></math>. We show that under natural assumptions, <span></span><math></math> is bounded only when <span></span><math></math> is very close to a twisted Dirichlet character <span></span><math></math>. We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when <span></span><math></math> is completely multiplicative and <span></span><math></math> is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70117","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615791","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
Polynomial-order oscillations in geometric discrepancy 几何差异中的多项式阶振荡
IF 0.6 3区 数学
Mathematika Pub Date : 2026-07-24 DOI: 10.1112/mtk.70121
Thomas Beretti
{"title":"Polynomial-order oscillations in geometric discrepancy","authors":"Thomas Beretti","doi":"10.1112/mtk.70121","DOIUrl":"https://doi.org/10.1112/mtk.70121","url":null,"abstract":"<p>Let <span></span><math></math> be a convex body, and for a positive integer <span></span><math></math>, let <span></span><math></math> be a configuration of <span></span><math></math> points in <span></span><math></math>. The discrepancy of <span></span><math></math> with respect to <span></span><math></math> is defined by\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615792","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
Galois representations are surjective for almost all Drinfeld modules 对于几乎所有的Drinfeld模，伽罗瓦表示都是满射的
IF 0.6 3区 数学
Mathematika Pub Date : 2026-07-23 DOI: 10.1112/mtk.70118
Anwesh Ray
{"title":"Galois representations are surjective for almost all Drinfeld modules","authors":"Anwesh Ray","doi":"10.1112/mtk.70118","DOIUrl":"https://doi.org/10.1112/mtk.70118","url":null,"abstract":"<p>This paper advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let <span></span><math></math> be the rational function field over a finite field. We establish that for Drinfeld modules of rank <span></span><math></math>, the <span></span><math></math>-adic Galois representation: <span></span><math></math> is surjective for a density 1 set of such modules. The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's uniformization theory and sieve methods.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615830","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
Galois representations are surjective for almost all Drinfeld modules 对于几乎所有的Drinfeld模，伽罗瓦表示都是满射的
IF 0.6 3区 数学
Mathematika Pub Date : 2026-07-23 DOI: 10.1112/mtk.70118
Anwesh Ray
{"title":"Galois representations are surjective for almost all Drinfeld modules","authors":"Anwesh Ray","doi":"10.1112/mtk.70118","DOIUrl":"https://doi.org/10.1112/mtk.70118","url":null,"abstract":"<p>This paper advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let <span></span><math></math> be the rational function field over a finite field. We establish that for Drinfeld modules of rank <span></span><math></math>, the <span></span><math></math>-adic Galois representation: <span></span><math></math> is surjective for a density 1 set of such modules. The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's uniformization theory and sieve methods.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615832","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
Matchable numbers 敌得过的数字
IF 0.6 3区 数学
Mathematika Pub Date : 2026-07-07 DOI: 10.1112/mtk.70116
Nathan McNew, Carl Pomerance
{"title":"Matchable numbers","authors":"Nathan McNew,&nbsp;Carl Pomerance","doi":"10.1112/mtk.70116","DOIUrl":"10.1112/mtk.70116","url":null,"abstract":"<p>We say a natural number <span></span><math></math> is matchable if there is a bijection from the set of <span></span><math></math> divisors of <span></span><math></math> to the set <span></span><math></math>, where corresponding numbers are relatively prime. We show that the set of matchable numbers has an asymptotic density, which we compute, and we show that every squarefree number is matchable. We also present some related unsolved problems.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148462775","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
Gluing posets and the dichotomy of poset saturation numbers 胶合偏序集与偏序集饱和数的二分法
IF 0.6 3区 数学
Mathematika Pub Date : 2026-06-26 DOI: 10.1112/mtk.70115
Maria-Romina Ivan, Sean Jaffe
{"title":"Gluing posets and the dichotomy of poset saturation numbers","authors":"Maria-Romina Ivan,&nbsp;Sean Jaffe","doi":"10.1112/mtk.70115","DOIUrl":"https://doi.org/10.1112/mtk.70115","url":null,"abstract":"<p>Given a finite poset <span></span><math></math>, we say that a family <span></span><math></math> of subsets of <span></span><math></math> is <span></span><math></math>-saturated if <span></span><math></math> does not contain an induced copy of <span></span><math></math>, but adding any other set to <span></span><math></math> creates an induced copy of <span></span><math></math>. The saturation number of <span></span><math></math> is the size of the smallest <span></span><math></math>-saturated family with ground set <span></span><math></math>. The saturation number for posets is known to exhibit a dichotomy: it is either bounded or it has a growth rate of at least <span></span><math></math>. Determining which posets have bounded saturation number is a major open problem. In this paper we consider a ‘gluing’ operation, formed from two finite posets <span></span><math></math> and <span></span><math></math> by setting all elements of <span></span><math></math> to be below all elements of <span></span><math></math>. We show that (under some mild assumptions) this operation preserves bounded and unbounded saturation number. This is the first such ‘new from old’ poset construction to be found. As an application, we show that for any poset <span></span><math></math> one may add at most 3 elements to <span></span><math></math> to obtain a poset whose saturation number growth is at most linear: this may be viewed as a step towards the other major open problem in the area, namely the conjecture that every finite poset has growth that is at most linear. We also consider the poset equivalent of weak saturation for graphs: for each finite poset <span></span><math></math>, we determine exactly the minimum size of a percolating family for <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2026-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324945","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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