{"title":"Uniform bounds for the density in Artin's conjecture on primitive roots","authors":"Antonella Perucca, Igor E. Shparlinski","doi":"10.1112/blms.70011","DOIUrl":"https://doi.org/10.1112/blms.70011","url":null,"abstract":"<p>We consider Artin's conjecture on primitive roots over a number field <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>, reducing an algebraic number <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mi>K</mi>\u0000 <mo>×</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$alpha in K^times$</annotation>\u0000 </semantics></math>. Under the Generalised Riemann Hypothesis, there is a density <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dens</mo>\u0000 <mo>(</mo>\u0000 <mi>α</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{dens}(alpha)$</annotation>\u0000 </semantics></math> counting the proportion of the primes of <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math> for which <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math> is a primitive root. This density <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dens</mo>\u0000 <mo>(</mo>\u0000 <mi>α</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{dens}(alpha)$</annotation>\u0000 </semantics></math> is a rational multiple of an Artin constant <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>(</mo>\u0000 <mi>τ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$A(tau)$</annotation>\u0000 </semantics></math> that depends on the largest integer <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>τ</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$tau geqslant 1$</annotation>\u0000 </semantics></math> such that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mfenced>\u0000 <msup>\u0000 <mi>K</mi>\u0000 <mo>×</mo>\u0000 </msup>\u0000 </mfenced>\u0000 <mi>τ</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$alpha in {left(K^ti","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"978-991"},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Embedding finitely presented self-similar groups into finitely presented simple groups","authors":"Matthew C. B. Zaremsky","doi":"10.1112/blms.70022","DOIUrl":"https://doi.org/10.1112/blms.70022","url":null,"abstract":"<p>We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone–Higman conjecture. The simple groups in question are certain commutator subgroups of Röver–Nekrashevych groups, and the difficulty lies in the fact that even if a Röver–Nekrashevych group is finitely presented, its commutator subgroup might not be. We also discuss a general example involving matrix groups over certain rings, which in particular establishes that every finitely generated subgroup of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mo>GL</mo>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Q</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$operatorname{GL}_n(mathbb {Q})$</annotation>\u0000 </semantics></math> satisfies the Boone–Higman conjecture.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1150-1159"},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Shi variety corresponding to an affine Weyl group","authors":"Nathan Chapelier-Laget","doi":"10.1112/blms.70007","DOIUrl":"https://doi.org/10.1112/blms.70007","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>W</mi>\u0000 <annotation>$W$</annotation>\u0000 </semantics></math> be an irreducible Weyl group and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math> its affine Weyl group. In this article we show that there exists a bijection between <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math> and the integral points of an affine variety, denoted <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>X</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$widehat{X}_{W_a}$</annotation>\u0000 </semantics></math>, which we call the Shi variety of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math>. In order to do so, we use Jian-Yi Shi's characterization of alcoves in affine Weyl groups. We then study this variety further. We introduce a new representation of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math>, called the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Φ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <annotation>$Phi ^+$</annotation>\u0000 </semantics></math>-representation, and we highlight combinatorial and geometrical properties of the irreducible components of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>X</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$widehat{X}_{W_a}$</annotation>\u0000 </semantics></math> via this representation. We also show how the components are related to a fundamental parallelepiped <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>P</mi>\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"913-940"},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Separability properties of higher rank GBS groups","authors":"Jone Lopez de Gamiz Zearra, Sam Shepherd","doi":"10.1112/blms.70024","DOIUrl":"https://doi.org/10.1112/blms.70024","url":null,"abstract":"<p>A rank <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> generalized Baumslag–Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {Z}^n$</annotation>\u0000 </semantics></math>. In this paper, we classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable, and cyclic subgroup separable.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1171-1194"},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Locally constant fibrations and positivity of curvature","authors":"Niklas Müller","doi":"10.1112/blms.70012","DOIUrl":"https://doi.org/10.1112/blms.70012","url":null,"abstract":"<p>Up to finite étale cover, any smooth complex projective variety <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> with nef anti-canonical bundle is a holomorphic fibre bundle over a smooth projective variety with trivial canonical class (<i>K</i>-trivial variety for short) with locally constant transition functions. We show that this result is optimal by proving that any projective fibre bundle with locally constant transition functions over a <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-trivial variety has a nef anti-canonical bundle. Moreover, we complement some results on the structure theory of varieties whose tangent bundle admits a singular Hermitian metric of positive curvature.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1005-1025"},"PeriodicalIF":0.8,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836162","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":"On Robin problems with infinite-point BCs","authors":"Chiun-Chang Lee","doi":"10.1112/blms.70018","DOIUrl":"https://doi.org/10.1112/blms.70018","url":null,"abstract":"<p>We study semilinear equations with Robin-type infinite-point boundary conditions, where the boundary conditions depend nonlocally on the solution at infinitely many interior points. To overcome the difficulties arising from the lack of guaranteed variational structure in these models, we establish mappings that correspond to the boundary conditions. Through a combination of asymptotic analysis and fixed-point arguments, we prove the existence of solutions under specific assumptions. This approach is novel in related research.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1093-1117"},"PeriodicalIF":0.8,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836161","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":"The homological spectrum via definable subcategories","authors":"Isaac Bird, Jordan Williamson","doi":"10.1112/blms.70014","DOIUrl":"https://doi.org/10.1112/blms.70014","url":null,"abstract":"<p>We develop an alternative approach to the homological spectrum of a tensor-triangulated category through the lens of definable subcategories. This culminates in a proof that the homological spectrum is homeomorphic to a quotient of the Ziegler spectrum. Along the way, we characterise injective objects in homological residue fields in terms of the definable subcategory corresponding to a given homological prime. We use these results to give a purity perspective on the relationship between the homological and Balmer spectrum.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1040-1064"},"PeriodicalIF":0.8,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835852","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":"Sometimes tame, sometimes wild: Weak continuity","authors":"Sam Sanders","doi":"10.1112/blms.70019","DOIUrl":"https://doi.org/10.1112/blms.70019","url":null,"abstract":"<p>Continuity is one of the most central notions in mathematics, physics and computer science. An interesting associated topic is <i>decompositions of continuity</i>, where continuity is shown to be equivalent to the combination of two or more <i>weak continuity</i> notions. In this paper, we study the logical properties of basic theorems about weakly continuous functions, like the supremum principle for the unit interval. We establish that most weak continuity notions are as tame as continuity, that is, the supremum principle can be proved from the relatively weak <i>arithmetical comprehension axiom</i> only. By contrast, for seven ‘wild’ weak continuity notions, the associated supremum principle yields rather strong axioms, including Feferman's projection principle, full second-order arithmetic or Kleene's associated quantifier <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mo>∃</mo>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(exists ^{3})$</annotation>\u0000 </semantics></math>. Working in Kohlenbach's higher-order <i>Reverse Mathematics</i>, we also obtain elegant equivalences in various cases and obtain similar results for for example, Riemann integration. We believe these results to be of interest to mainstream mathematics as they cast new light on the distinction of ‘ordinary mathematics’ versus ‘foundations of mathematics/set theory’.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1324-1346"},"PeriodicalIF":0.8,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David Gonzalez, Matthew Harrison-Trainor, Meng-Che “Turbo” Ho
{"title":"Scott analysis, linear orders, and almost periodic functions","authors":"David Gonzalez, Matthew Harrison-Trainor, Meng-Che “Turbo” Ho","doi":"10.1112/blms.70020","DOIUrl":"https://doi.org/10.1112/blms.70020","url":null,"abstract":"<p>Given a countable structure, the Scott complexity measures the difficulty of characterizing the structure up to isomorphism. In this paper, we consider the Scott complexity of linear orders. For any limit ordinal <span></span><math>\u0000 <semantics>\u0000 <mi>λ</mi>\u0000 <annotation>$lambda$</annotation>\u0000 </semantics></math>, we construct a linear order <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>λ</mi>\u0000 </msub>\u0000 <annotation>$L_lambda$</annotation>\u0000 </semantics></math> whose Scott complexity is <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$Sigma _{lambda +1}$</annotation>\u0000 </semantics></math>. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$Sigma _{lambda +1}$</annotation>\u0000 </semantics></math>, and our construction gives new examples, for example, rigid structures, of this complexity. Moreover, we can construct the linear orders <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>λ</mi>\u0000 </msub>\u0000 <annotation>$L_lambda$</annotation>\u0000 </semantics></math> so that not only does <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>λ</mi>\u0000 </msub>\u0000 <annotation>$L_lambda$</annotation>\u0000 </semantics></math> have Scott complexity <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$Sigma _{lambda +1}$</annotation>\u0000 </semantics></math>, but there are continuum-many structures <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 <msub>\u0000 <mo>≡</mo>\u0000 <mi>λ</mi>\u0000 </msub>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>λ</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$M equiv","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1118-1139"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836014","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}
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