Let be a smooth irreducible quasi-projective algebraic variety over a number field . Suppose is equipped with a -adic étale local system compatible with an admissible graded-polarized variation of mixed Hodge structures on the complex analytification of . We prove that the -integral points in are covered by subpolynomially many geometrically irreducible -subvarieties, each lying in a fiber of the mixed period mapping arising from the variation of mixed Hodge structures. This is based on recent works by Brunebarbe–Maculan and Ellenberg–Lawrence–Venkatesh. As an application, we prove that there are subpolynomially many -integral Laurent polynomials with fixed reflexive Newton polyhedron and fixed non-zero principal -determinant. Our results answer a question asked by Ellenberg–Lawrence–Venkatesh.
{"title":"Arithmetic sparsity in mixed Hodge settings","authors":"Kenneth Chung Tak Chiu","doi":"10.1112/blms.70166","DOIUrl":"https://doi.org/10.1112/blms.70166","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> be a smooth irreducible quasi-projective algebraic variety over a number field <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>. Suppose <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> is equipped with a <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-adic étale local system compatible with an admissible graded-polarized variation of mixed Hodge structures on the complex analytification of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>X</mi>\u0000 <mo>C</mo>\u0000 </msub>\u0000 <annotation>$X_{operatorname{mathbb {C}}}$</annotation>\u0000 </semantics></math>. We prove that the <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>-integral points in <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> are covered by subpolynomially many geometrically irreducible <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-subvarieties, each lying in a fiber of the mixed period mapping arising from the variation of mixed Hodge structures. This is based on recent works by Brunebarbe–Maculan and Ellenberg–Lawrence–Venkatesh. As an application, we prove that there are subpolynomially many <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>-integral Laurent polynomials with fixed reflexive Newton polyhedron <span></span><math>\u0000 <semantics>\u0000 <mi>Δ</mi>\u0000 <annotation>$Delta$</annotation>\u0000 </semantics></math> and fixed non-zero principal <span></span><math>\u0000 <semantics>\u0000 <mi>Δ</mi>\u0000 <annotation>$Delta$</annotation>\u0000 </semantics></math>-determinant. Our results answer a question asked by Ellenberg–Lawrence–Venkatesh.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 11","pages":"3511-3521"},"PeriodicalIF":0.9,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486966","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}
In this informal expository note, we quickly introduce and survey compactifications of strata of holomorphic 1-forms on Riemann surfaces, that is, spaces of translation surfaces. In the last decade, several of these have been constructed, studied, and successfully applied to problems. We discuss relations between their definitions and properties, focusing on the different notions of convergence from a flat geometric perspective.
{"title":"Compactifications of strata of differentials","authors":"Benjamin Dozier","doi":"10.1112/blms.70153","DOIUrl":"https://doi.org/10.1112/blms.70153","url":null,"abstract":"<p>In this informal expository note, we quickly introduce and survey compactifications of strata of holomorphic 1-forms on Riemann surfaces, that is, spaces of translation surfaces. In the last decade, several of these have been constructed, studied, and successfully applied to problems. We discuss relations between their definitions and properties, focusing on the different notions of convergence from a flat geometric perspective.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 12","pages":"3621-3636"},"PeriodicalIF":0.9,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70153","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145848354","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}
In the semisimple case, we derive (asymptotic) formulas for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.
在半简单情况下，我们导出了图/插值类中生成对象的张量幂和数增长率的（渐近）公式。
{"title":"Growth problems in diagram categories","authors":"Jonathan Gruber, Daniel Tubbenhauer","doi":"10.1112/blms.70163","DOIUrl":"https://doi.org/10.1112/blms.70163","url":null,"abstract":"<p>In the semisimple case, we derive (asymptotic) formulas for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 11","pages":"3454-3469"},"PeriodicalIF":0.9,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70163","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145487093","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}
We prove that (under appropriate orientation assumptions), the action of a Hamiltonian homeomorphism on the cohomology of a relatively exact Lagrangian fixed by is the identity. This extends results of Hu–Lalonde–Leclercq [Geom. Topol. 15 (2011), no. 3, 1617–1650] and the author [Selecta Math. (N.S.) 30 (2024), no. 2, Paper No. 21, 53] in the setting of Hamiltonian diffeomorphisms. We also prove a similar result regarding the action of on relative cohomology.
{"title":"C\u0000 0\u0000 \u0000 $C^0$\u0000 Lagrangian monodromy","authors":"Noah Porcelli","doi":"10.1112/blms.70162","DOIUrl":"https://doi.org/10.1112/blms.70162","url":null,"abstract":"<p>We prove that (under appropriate orientation assumptions), the action of a Hamiltonian homeomorphism <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math> on the cohomology of a relatively exact Lagrangian fixed by <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math> is the identity. This extends results of Hu–Lalonde–Leclercq [Geom. Topol. <b>15</b> (2011), no. 3, 1617–1650] and the author [Selecta Math. (N.S.) <b>30</b> (2024), no. 2, Paper No. 21, 53] in the setting of Hamiltonian diffeomorphisms. We also prove a similar result regarding the action of <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math> on relative cohomology.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 11","pages":"3441-3453"},"PeriodicalIF":0.9,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70162","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145487104","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}
We use high girth, high chromatic number hypergraphs to show that there are finite models of the equational theory of the semiring of non-negative integers whose equational theory has no finite axiomatisation, and show this also holds if factorial, fixed base exponentiation and operations for binomial coefficients are adjoined. We also derive the decidability of the equational logical entailment operator for antecedents true on by way of a form of the finite model property. Two appendices contain additional basic development of combinatorial operations. Amongst the observations are an eventual dominance well-ordering of combinatorial functions and consequent representation of the ordinal in terms of factorial functions; the equivalence of the equational logic of combinatorial algebra over the natural numbers and over the positive reals; and a candidate list of elementary axioms.
{"title":"Finite models for positive combinatorial and exponential algebra","authors":"Tumadhir Alsulami, Marcel Jackson","doi":"10.1112/blms.70158","DOIUrl":"https://doi.org/10.1112/blms.70158","url":null,"abstract":"<p>We use high girth, high chromatic number hypergraphs to show that there are finite models of the equational theory of the semiring of non-negative integers whose equational theory has no finite axiomatisation, and show this also holds if factorial, fixed base exponentiation and operations for binomial coefficients are adjoined. We also derive the decidability of the equational logical entailment operator <span></span><math>\u0000 <semantics>\u0000 <mo>⊢</mo>\u0000 <annotation>$vdash$</annotation>\u0000 </semantics></math> for antecedents true on <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$mathbb {N}$</annotation>\u0000 </semantics></math> by way of a form of the finite model property. Two appendices contain additional basic development of combinatorial operations. Amongst the observations are an eventual dominance well-ordering of combinatorial functions and consequent representation of the ordinal <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ε</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$epsilon _0$</annotation>\u0000 </semantics></math> in terms of factorial functions; the equivalence of the equational logic of combinatorial algebra over the natural numbers and over the positive reals; and a candidate list of elementary axioms.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 11","pages":"3380-3400"},"PeriodicalIF":0.9,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70158","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145487097","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}
Josef Greilhuber, Carl Schildkraut, Jonathan Tidor
<p>For <span></span><math> <semantics> <mrow> <mi>d</mi> <mo>⩾</mo> <mn>2</mn> </mrow> <annotation>$dgeqslant 2$</annotation> </semantics></math> and any norm on <span></span><math> <semantics> <msup> <mi>R</mi> <mi>d</mi> </msup> <annotation>$mathbb {R}^d$</annotation> </semantics></math>, we prove that there exists a set of <span></span><math> <semantics> <mi>n</mi> <annotation>$n$</annotation> </semantics></math> points that spans at least <span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mstyle> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> </mstyle> <mo>−</mo> <mi>o</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>n</mi> <msub> <mi>log</mi> <mn>2</mn> </msub> <mi>n</mi> </mrow> <annotation>$(tfrac{d}{2}-o(1))nlog _2n$</annotation> </semantics></math> unit distances under this norm for every <span></span><math> <semantics> <mi>n</mi> <annotation>$n$</annotation> </semantics></math>. This matches the upper bound recently proved by Alon, Bucić, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for <span></span><math> <semantics> <mrow> <mi>d</mi> <mo>⩾</mo> <mn>3</mn> </mrow> <annotation>$dgeqslant 3$</annotation> </semantics></math> and a typical norm on <span></span><math> <semantics> <msup> <mi>R</mi> <mi>d</mi> </msup> <annotation>$mathbb {R}^d$</annotation> </semantics></math>, the unit distance graph of this norm contains a copy of <span></span><math> <semantics> <msub> <mi>K</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <annotation>$K_{d,m}$</annotation> </semantics></math> for all <span></span><math> <semantics> <mi>m</mi> <annot
对于d小于2 $dgeqslant 2$和R d $mathbb {R}^d$上的任何规范，我们证明了存在一个n个$n$点的集合，它张成至少（d 2−o (1)）) n log 2 n $(tfrac{d}{2}-o(1))nlog _2n$在这个范数下的单位距离对于每一个n $n$。这与最近由Alon， buciki和Sauermann证明的典型范数（即，位于一个趋同集中的范数）的上界相匹配。我们还显示，对于d小于3 $dgeqslant 3$和R d $mathbb {R}^d$上的典型范数，该范数的单位距离图包含K d的副本，M $K_{d,m}$代表所有M $m$。
{"title":"More unit distances in arbitrary norms","authors":"Josef Greilhuber, Carl Schildkraut, Jonathan Tidor","doi":"10.1112/blms.70133","DOIUrl":"10.1112/blms.70133","url":null,"abstract":"<p>For <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 2$</annotation>\u0000 </semantics></math> and any norm on <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^d$</annotation>\u0000 </semantics></math>, we prove that there exists a set of <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> points that spans at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mstyle>\u0000 <mfrac>\u0000 <mi>d</mi>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mstyle>\u0000 <mo>−</mo>\u0000 <mi>o</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 <msub>\u0000 <mi>log</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$(tfrac{d}{2}-o(1))nlog _2n$</annotation>\u0000 </semantics></math> unit distances under this norm for every <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>. This matches the upper bound recently proved by Alon, Bucić, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 3$</annotation>\u0000 </semantics></math> and a typical norm on <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^d$</annotation>\u0000 </semantics></math>, the unit distance graph of this norm contains a copy of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>,</mo>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$K_{d,m}$</annotation>\u0000 </semantics></math> for all <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annot","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 9","pages":"2885-2901"},"PeriodicalIF":0.9,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145022368","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}
<p>We obtain restrictions on units of even order in the integral group ring <span></span><math> <semantics> <mrow> <mi>Z</mi> <mi>G</mi> </mrow> <annotation>$mathbb {Z}G$</annotation> </semantics></math> of a finite group <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> by studying their actions on the reductions modulo 4 of lattices over the 2-adic group ring <span></span><math> <semantics> <mrow> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mi>G</mi> </mrow> <annotation>$mathbb {Z}_2G$</annotation> </semantics></math>. This improves the “lattice method” which considers reductions modulo primes <span></span><math> <semantics> <mi>p</mi> <annotation>$p$</annotation> </semantics></math>, but is of limited use for <span></span><math> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> <annotation>$p=2$</annotation> </semantics></math> essentially due to the fact that <span></span><math> <semantics> <mrow> <mn>1</mn> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mspace></mspace> <mo>(</mo> <mi>mod</mi> <mspace></mspace> <mn>2</mn> <mo>)</mo> </mrow> <annotation>$1equiv -1 (text{mod }2)$</annotation> </semantics></math>. Our methods yield results in cases where <span></span><math> <semantics> <mrow> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mi>G</mi> </mrow> <annotation>$mathbb {Z}_2 G$</annotation> </semantics></math> has blocks, whose defect groups are Klein four groups or dihedral groups of order 8. This allows us to disprove the existence of units of order <span></span><math> <semantics> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <annotation>$2p$</annotation> </semantics></math> for almost simple groups with socle <span></span><math> <semantics> <mrow> <mo>PSL</mo>
{"title":"Units in group rings and blocks of Klein four or dihedral defect","authors":"Florian Eisele, Leo Margolis","doi":"10.1112/blms.70164","DOIUrl":"https://doi.org/10.1112/blms.70164","url":null,"abstract":"<p>We obtain restrictions on units of even order in the integral group ring <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Z</mi>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$mathbb {Z}G$</annotation>\u0000 </semantics></math> of a finite group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> by studying their actions on the reductions modulo 4 of lattices over the 2-adic group ring <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$mathbb {Z}_2G$</annotation>\u0000 </semantics></math>. This improves the “lattice method” which considers reductions modulo primes <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>, but is of limited use for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p=2$</annotation>\u0000 </semantics></math> essentially due to the fact that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>≡</mo>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mspace></mspace>\u0000 <mo>(</mo>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$1equiv -1 (text{mod }2)$</annotation>\u0000 </semantics></math>. Our methods yield results in cases where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$mathbb {Z}_2 G$</annotation>\u0000 </semantics></math> has blocks, whose defect groups are Klein four groups or dihedral groups of order 8. This allows us to disprove the existence of units of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <annotation>$2p$</annotation>\u0000 </semantics></math> for almost simple groups with socle <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>PSL</mo>\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 11","pages":"3470-3489"},"PeriodicalIF":0.9,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70164","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486926","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}
<p>The normal covering number <span></span><math> <semantics> <mrow> <mi>γ</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <annotation>$gamma (G)$</annotation> </semantics></math> of a noncyclic group <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for <span></span><math> <semantics> <mrow> <mi>γ</mi> <mo>(</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <annotation>$gamma (S_n)$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>γ</mi> <mo>(</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <annotation>$gamma (A_n)$</annotation> </semantics></math> depending on the arithmetic structure of <span></span><math> <semantics> <mi>n</mi> <annotation>$n$</annotation> </semantics></math>. In particular we determine the limsups over <span></span><math> <semantics> <mrow> <mi>γ</mi> <mo>(</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>/</mo> <mi>n</mi> </mrow> <annotation>$gamma (S_n) / n$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>γ</mi> <mo>(</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>/</mo> <mi>n</mi> </mrow> <annotation>$gamma (A_n) / n$</annotation> </semantics></math> over the sequences of even and odd integers, as well as the liminf of <span></span><math> <semantics> <mrow> <mi>γ</mi> <mo>(</mo> <msub> <mi>S</mi> <mi>n</mi> </ms
{"title":"Normal covering numbers for \u0000 \u0000 \u0000 S\u0000 n\u0000 \u0000 $S_n$\u0000 and \u0000 \u0000 \u0000 A\u0000 n\u0000 \u0000 $A_n$\u0000 and additive combinatorics","authors":"Sean Eberhard, Connor Mellon","doi":"10.1112/blms.70154","DOIUrl":"https://doi.org/10.1112/blms.70154","url":null,"abstract":"<p>The normal covering number <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$gamma (G)$</annotation>\u0000 </semantics></math> of a noncyclic group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$gamma (S_n)$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$gamma (A_n)$</annotation>\u0000 </semantics></math> depending on the arithmetic structure of <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>. In particular we determine the limsups over <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 <mo>/</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$gamma (S_n) / n$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 <mo>/</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$gamma (A_n) / n$</annotation>\u0000 </semantics></math> over the sequences of even and odd integers, as well as the liminf of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </ms","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 11","pages":"3307-3325"},"PeriodicalIF":0.9,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70154","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145487073","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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