{"title":"Inhomogeneous Khintchine–Groshev theorem without monotonicity","authors":"Seongmin Kim","doi":"10.1112/blms.70114","DOIUrl":"10.1112/blms.70114","url":null,"abstract":"<p>The Khintchine–Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of <span></span><math>\u0000 <semantics>\u0000 <mi>ψ</mi>\u0000 <annotation>$psi$</annotation>\u0000 </semantics></math>-approximable numbers, given a monotonic function <span></span><math>\u0000 <semantics>\u0000 <mi>ψ</mi>\u0000 <annotation>$psi$</annotation>\u0000 </semantics></math>. Allen and Ramírez removed the monotonicity condition from the inhomogeneous Khintchine–Groshev theorem for cases with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mi>m</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$nmgeqslant 3$</annotation>\u0000 </semantics></math> and conjectured that it also holds for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mi>m</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$nm=2$</annotation>\u0000 </semantics></math>. In this paper, we prove this conjecture in the case of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(n,m)=(2,1)$</annotation>\u0000 </semantics></math>. We also prove it for the case of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(n,m)=(1,2)$</annotation>\u0000 </semantics></math> with a rational inhomogeneous parameter.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 9","pages":"2639-2657"},"PeriodicalIF":0.9,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70114","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145021917","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 profinite groups with the Magnus Property","authors":"Claude Marion, Pavel Zalesskii","doi":"10.1112/blms.70107","DOIUrl":"10.1112/blms.70107","url":null,"abstract":"<p>A group is said to have the Magnus Property (MP) if whenever two elements have the same normal closure then they are conjugate or inverse-conjugate. We show that a profinite MP group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is prosolvable and any quotient of it is again MP. As corollaries, we obtain that a prime divisor of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>G</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <annotation>$|G|$</annotation>\u0000 </semantics></math> is 2, 3, 5 or 7, and the second derived subgroup of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is pronilpotent. We also show that the inverse limit of an inverse system of profinite MP groups is again MP. Finally when <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is finitely generated, we establish that <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> must in fact be finite.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2489-2497"},"PeriodicalIF":0.9,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144815044","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":"Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant","authors":"Leandro F. Pessoa, Erisvaldo Véras, Bruno Vieira","doi":"10.1112/blms.70118","DOIUrl":"10.1112/blms.70118","url":null,"abstract":"<p>We prove area estimates for stable capillary <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>c</mi>\u0000 <mi>m</mi>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$cmc$</annotation>\u0000 </semantics></math> (minimal) hypersurfaces <span></span><math>\u0000 <semantics>\u0000 <mi>Σ</mi>\u0000 <annotation>$Sigma$</annotation>\u0000 </semantics></math> with nonpositive Yamabe invariant that are properly immersed in a Riemannian <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-dimensional manifold <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> with scalar curvature <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>M</mi>\u0000 </msup>\u0000 <annotation>$R^M$</annotation>\u0000 </semantics></math> and mean curvature of the boundary <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mrow>\u0000 <mi>∂</mi>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$H^{partial M}$</annotation>\u0000 </semantics></math> bounded from below. We also prove a local rigidity result in the case <span></span><math>\u0000 <semantics>\u0000 <mi>Σ</mi>\u0000 <annotation>$Sigma$</annotation>\u0000 </semantics></math> is embedded and <span></span><math>\u0000 <semantics>\u0000 <mi>J</mi>\u0000 <annotation>$mathcal {J}$</annotation>\u0000 </semantics></math>-energy-minimizing. In this case, we show that <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> locally splits along <span></span><math>\u0000 <semantics>\u0000 <mi>Σ</mi>\u0000 <annotation>$Sigma$</annotation>\u0000 </semantics></math> and is isometric to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mo>−</mo>\u0000 <mi>ε</mi>\u0000 <mo>,</mo>\u0000 <mi>ε</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>×</mo>\u0000 <mi>Σ</mi>\u0000 <mo>,</mo>\u0000 <mi>d</mi>\u0000 <msup>\u0000 <mi>t</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>e</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 <mi>H</m","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 9","pages":"2708-2722"},"PeriodicalIF":0.9,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145022116","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}
Naihuan Jing, Zhijun Li, Xinyu Pan, Danxia Wang, Chang Ye
{"title":"Skew odd orthogonal characters and interpolating Schur polynomials","authors":"Naihuan Jing, Zhijun Li, Xinyu Pan, Danxia Wang, Chang Ye","doi":"10.1112/blms.70109","DOIUrl":"10.1112/blms.70109","url":null,"abstract":"<p>We introduce two vertex operators to realize skew odd orthogonal characters <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <msub>\u0000 <mi>o</mi>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>/</mo>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>x</mi>\u0000 <mo>±</mo>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$so_{lambda /mu }(mathbf {x}^{pm })$</annotation>\u0000 </semantics></math> and derive the Cauchy identity for the skew characters via the Toeplitz–Hankel-type determinant as an analog of Schur functions. The method also gives new proofs of the Jacobi–Trudi identity and Gelfand–Tsetlin patterns for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <msub>\u0000 <mi>o</mi>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>/</mo>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>x</mi>\u0000 <mo>±</mo>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$so_{lambda /mu }(mathbf {x}^{pm })$</annotation>\u0000 </semantics></math>. Moreover, combining the vertex operators related to characters of types <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>,</mo>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation>$C,D$</annotation>\u0000 </semantics></math> (Baker, <i>J. Phys. A</i>. <b>29</b>(12) (1996), 3099–3117; Jing and Nie, <i>Ann. Combin</i>. <b>19</b> (2015), 427–442) and the new vertex operators related to <span></span><math>\u0000 <semantics>\u0000 <mi>B</mi>\u0000 <annotation>$B$</annotation>\u0000 </semantics></math>-type characters, we obtain three families of symmetric polynomials that interpolate among characters of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>SO</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>C</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2509-2530"},"PeriodicalIF":0.9,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144814923","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 Poincaré polynomials for plane curves with quasi-homogeneous singularities","authors":"Piotr Pokora","doi":"10.1112/blms.70112","DOIUrl":"10.1112/blms.70112","url":null,"abstract":"<p>We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincaré polynomial. We prove a Terao-type factorization statement on the splitting of such a polynomial over the rationals under the assumption that our conic-line arrangements are free and admit ordinary quasi-homogeneous singularities. Then we focus on the so-called <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>-arrangements in the plane. In particular, we provide a combinatorial constraint for free <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>-arrangements admitting ordinary quasi-homogeneous singularities.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2549-2560"},"PeriodicalIF":0.9,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144815220","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":"Construction of a curved Kakeya set","authors":"Tongou Yang, Yue Zhong","doi":"10.1112/blms.70110","DOIUrl":"10.1112/blms.70110","url":null,"abstract":"<p>We construct a compact set in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math> of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math>-<span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>q</mi>\u0000 </msup>\u0000 <annotation>$L^q$</annotation>\u0000 </semantics></math> norm of the corresponding maximal operator for a range of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation>$p,q$</annotation>\u0000 </semantics></math>. Moreover, our construction can be generalised from parabolas to a family of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$C^2$</annotation>\u0000 </semantics></math> curves satisfying suitable curvature conditions.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2531-2548"},"PeriodicalIF":0.9,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144815219","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":"Corrigendum: Graph bundles and Ricci-flatness","authors":"Wenbo Li, Shiping Liu","doi":"10.1112/blms.70111","DOIUrl":"10.1112/blms.70111","url":null,"abstract":"<p>This note is a corrigendum to the authors' paper “Graph bundles and Ricci–flatness, Bulletin of the London Mathematical Society, 56(2), pp. 523–535, 2024”.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1921-1922"},"PeriodicalIF":0.9,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70111","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144245168","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}
Lorena Aguirre Salazar, Xin Yang Lu, Jun-cheng Wei
{"title":"An Ohta–Kawasaki model set on the space","authors":"Lorena Aguirre Salazar, Xin Yang Lu, Jun-cheng Wei","doi":"10.1112/blms.70105","DOIUrl":"10.1112/blms.70105","url":null,"abstract":"<p>We examine a nonlocal diffuse interface energy with Coulomb repulsion in three dimensions inspired by the Thomas–Fermi–Dirac–von Weizsäcker, and the Ohta–Kawasaki models. We consider the corresponding mass-constrained variational problem and show the existence of minimizers for small masses, and the absence of minimizers for large masses.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2462-2476"},"PeriodicalIF":0.9,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144815248","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":"Thurston obstructions and tropical geometry","authors":"Rohini Ramadas","doi":"10.1112/blms.70102","DOIUrl":"10.1112/blms.70102","url":null,"abstract":"<p>We describe an application of tropical moduli spaces to complex dynamics. A post-critically finite branched covering <span></span><math>\u0000 <semantics>\u0000 <mi>φ</mi>\u0000 <annotation>$varphi$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$S^2$</annotation>\u0000 </semantics></math> induces a pullback map on the Teichmüller space of complex structures of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$S^2$</annotation>\u0000 </semantics></math>; this descends to an algebraic correspondence on the moduli space of point-configurations of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$mathbb {P}^1$</annotation>\u0000 </semantics></math>. We make a case for studying the action of the tropical moduli space correspondence by making explicit the connections between objects that have come up in one guise in tropical geometry and in another guise in complex dynamics. For example, a Thurston obstruction for <span></span><math>\u0000 <semantics>\u0000 <mi>φ</mi>\u0000 <annotation>$varphi$</annotation>\u0000 </semantics></math> corresponds to a ray that is fixed by the tropical moduli space correspondence, and scaled by a factor <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$geqslant 1$</annotation>\u0000 </semantics></math>. This article is intended to be accessible to algebraic and tropical geometers as well as to complex dynamicists.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2404-2428"},"PeriodicalIF":0.9,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70102","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144815157","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}