Bulletin of the London Mathematical Society最新文献

{"title":"Minimal varieties of graded PI-algebras over abelian groups","authors":"Sebastiano Argenti,&nbsp;Onofrio Mario Di Vincenzo","doi":"10.1112/blms.13064","DOIUrl":"https://doi.org/10.1112/blms.13064","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math> be a field of characteristic zero and <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> a finite abelian group. In this paper, we prove that an affine variety of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>-graded PI-algebras is minimal if and only if it is generated by a graded algebra <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>U</mi>\u0000 <mi>T</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mi>⋯</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mo>;</mo>\u0000 <mi>γ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$UT(A_1,dots,A_m;gamma)$</annotation>\u0000 </semantics></math> of upper block triangular matrices where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mi>⋯</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$A_1,dots,A_m$</annotation>\u0000 </semantics></math> are finite-dimensional <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>-simple algebras.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 7","pages":"2441-2459"},"PeriodicalIF":0.8,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141556688","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":"Weighted Alexandrov–Fenchel type inequalities for hypersurfaces in \u0000 \u0000 \u0000 R\u0000 n\u0000 \u0000 $mathbb {R}^n$","authors":"Jie Wu","doi":"10.1112/blms.13089","DOIUrl":"10.1112/blms.13089","url":null,"abstract":"<p>In this paper, we prove the following geometric inequalities in the Euclidean space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mspace></mspace>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathbb {R}^n (ngeqslant 3)$</annotation>\u0000 </semantics></math>, which are weighted Alexandrov–Fenchel type inequalities,\u0000\u0000 </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2634-2646"},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141099192","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":"Milnor–Wood inequality for klt varieties of general type and uniformization","authors":"Matteo Costantini,&nbsp;Daniel Greb","doi":"10.1112/blms.13071","DOIUrl":"https://doi.org/10.1112/blms.13071","url":null,"abstract":"<p>We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood-type inequality and we characterize the corresponding maximal representations.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 7","pages":"2552-2567"},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141556727","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":"Growth of products of subsets in finite simple groups","authors":"Daniele Dona,&nbsp;Attila Maróti,&nbsp;László Pyber","doi":"10.1112/blms.13093","DOIUrl":"https://doi.org/10.1112/blms.13093","url":null,"abstract":"<p>We prove that the product of a subset and a normal subset inside any finite simple non-abelian group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> grows rapidly. More precisely, if <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>B</mi>\u0000 <annotation>$B$</annotation>\u0000 </semantics></math> are two subsets with <span></span><math>\u0000 <semantics>\u0000 <mi>B</mi>\u0000 <annotation>$B$</annotation>\u0000 </semantics></math> normal and neither of them is too large inside <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, then <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>A</mi>\u0000 <mi>B</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mo>⩾</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>A</mi>\u0000 <mo>|</mo>\u0000 <mo>|</mo>\u0000 <mi>B</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mi>ε</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$|AB| geqslant |A||B|^{1-epsilon }$</annotation>\u0000 </semantics></math> where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$epsilon &amp;gt;0$</annotation>\u0000 </semantics></math> can be taken arbitrarily small. This is a somewhat surprising strengthening of a theorem of Liebeck, Schul, and Shalev.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2704-2710"},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968089","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":"Quantitative upper bounds related to an isogeny criterion for elliptic curves","authors":"Alina Carmen Cojocaru,&nbsp;Auden Hinz,&nbsp;Tian Wang","doi":"10.1112/blms.13091","DOIUrl":"https://doi.org/10.1112/blms.13091","url":null,"abstract":"&lt;p&gt;For &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$E_1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$E_2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; elliptic curves defined over a number field &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, without complex multiplication, we consider the function &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;${mathcal {F}}_{E_1, E_2}(x)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; counting nonzero prime ideals &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {p}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the ring of integers of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, of good reduction for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$E_1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$E_2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, of norm at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;annotation&gt;$x$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and for which the Frobenius fields &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;π&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msu","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2661-2679"},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13091","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968099","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":"Higher Morita–Tachikawa correspondence","authors":"Tiago Cruz","doi":"10.1112/blms.13090","DOIUrl":"https://doi.org/10.1112/blms.13090","url":null,"abstract":"<p>Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo-symmetric algebras. Explicitly, the Morita–Tachikawa correspondence describes that endomorphism algebras of generators–cogenerators over finite-dimensional algebras are exactly the finite-dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi-generators and quasi-cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita–Tachikawa correspondence that take into account relative dominant dimension with respect to a self-orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2647-2660"},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13090","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967415","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":"The isomorphism problem for oligomorphic groups with weak elimination of imaginaries","authors":"Gianluca Paolini","doi":"10.1112/blms.13086","DOIUrl":"https://doi.org/10.1112/blms.13086","url":null,"abstract":"<p>In Kechris et al. [J. Symb. Log. <b>83</b> (2018), no. 3, 1190–1203], it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries, this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Aut</mi>\u0000 <mo>(</mo>\u0000 <mi>GL</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{Aut}(mathrm{GL}(V))$</annotation>\u0000 </semantics></math> for any vector space <span></span><math>\u0000 <semantics>\u0000 <mi>V</mi>\u0000 <annotation>$V$</annotation>\u0000 </semantics></math> of dimension <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$aleph _0$</annotation>\u0000 </semantics></math> over a finite field, in affinity with the classical description for finite-dimensional spaces due to Schreier and van der Waerden.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2597-2614"},"PeriodicalIF":0.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968026","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":"Infinitely many Riemann surfaces with a transitive action on the Weierstrass points","authors":"Sebastián Reyes-Carocca,&nbsp;Pietro Speziali","doi":"10.1112/blms.13088","DOIUrl":"10.1112/blms.13088","url":null,"abstract":"<p>In this short note, we prove the existence of infinitely many pairwise nonisomorphic, non-hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also find all compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are simple.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2625-2633"},"PeriodicalIF":0.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141122706","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":"Stability and equivariant Gromov–Hausdorff convergence","authors":"Mohammad Alattar","doi":"10.1112/blms.13073","DOIUrl":"https://doi.org/10.1112/blms.13073","url":null,"abstract":"<p>We give applications of equivariant Gromov–Hausdorff convergence in various contexts. Namely, using equivariant Gromov–Hausdorff convergence, we prove a stability result in the setting of compact finite-dimensional Alexandrov spaces. Moreover, we introduce the notion of an <i>almost commutative diagram</i> and show that its use simplifies both exposition and argument.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2585-2596"},"PeriodicalIF":0.8,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13073","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967988","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":"Radical bound for Zaremba's conjecture","authors":"Nikita Shulga","doi":"10.1112/blms.13087","DOIUrl":"10.1112/blms.13087","url":null,"abstract":"&lt;p&gt;Famous Zaremba's conjecture (1971) states that for each positive integer &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$qgeqslant 2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, there exists a positive integer &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$1leqslant a &amp;lt;q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, coprime to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;annotation&gt;$q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, such that if you expand a fraction &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$a/q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; into a continued fraction &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$a/q=[a_1,ldots,a_n]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, all of the coefficients &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$a_i$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;’s are bounded by some absolute constant &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {k}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, independent of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;annotation&gt;$q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Zaremba conjectured that this should hold for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathfrak {k}=5$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/ma","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2615-2624"},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13087","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141126648","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}