Journal of the London Mathematical Society-Second Series最新文献

{"title":"Effective generic freeness and applications to local cohomology","authors":"Yairon Cid-Ruiz,&nbsp;Ilya Smirnov","doi":"10.1112/jlms.12995","DOIUrl":"https://doi.org/10.1112/jlms.12995","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> be a Noetherian domain and <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> be a finitely generated <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math>-algebra. We study several features regarding the generic freeness over <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> of an <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>-module. For an ideal <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>I</mi>\u0000 <mo>⊂</mo>\u0000 <mi>R</mi>\u0000 </mrow>\u0000 <annotation>$I subset R$</annotation>\u0000 </semantics></math>, we show that the local cohomology modules <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>H</mi>\u0000 <mi>I</mi>\u0000 <mi>i</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$normalfont text{H}_I^i(R)$</annotation>\u0000 </semantics></math> are generically free over <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> under certain settings where <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> is a smooth <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math>-algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> of a finitely generated <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>-module.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142273278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Time-periodic solutions to heated ferrofluid flow models","authors":"Kamel Hamdache,&nbsp;Djamila Hamroun,&nbsp;Basma Jaffal-Mourtada","doi":"10.1112/jlms.12990","DOIUrl":"https://doi.org/10.1112/jlms.12990","url":null,"abstract":"<p>In this work, we prove the existence of time-periodic solutions to a model describing a ferrofluid flow heated from below. Navier–Stokes equations satisfied by the fluid velocity are coupled to the temperature equation and the magnetostatic equation satisfied by the magnetic potential. The magnetization is assumed to be parallel to the magnetic field and is given by a nonlinear magnetization law generalizing the Langevin law. The proof is based on a semi-Galerkin approximation and regularization methods together with the fixed point method.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142273003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fullness of \u0000 \u0000 q\u0000 $q$\u0000 -Araki-Woods factors","authors":"Manish Kumar,&nbsp;Simeng Wang","doi":"10.1112/jlms.12989","DOIUrl":"https://doi.org/10.1112/jlms.12989","url":null,"abstract":"<p>The <span></span><math>\u0000 <semantics>\u0000 <mi>q</mi>\u0000 <annotation>$q$</annotation>\u0000 </semantics></math>-Araki-Woods factor associated to a group of orthogonal transformations on a real separable Hilbert space <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>R</mi>\u0000 </msub>\u0000 <annotation>$mathsf {H}_{mathbb {R}}$</annotation>\u0000 </semantics></math> is full as soon as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dim</mo>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>R</mi>\u0000 </msub>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$dim mathsf {H}_{mathbb {R}}geqslant 2$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142244857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lattice reduced and complete convex bodies","authors":"Giulia Codenotti,&nbsp;Ansgar Freyer","doi":"10.1112/jlms.12982","DOIUrl":"https://doi.org/10.1112/jlms.12982","url":null,"abstract":"&lt;p&gt;The purpose of this paper is to study convex bodies &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;annotation&gt;$C$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for which there exists no convex body &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mi&gt;⊊&lt;/mi&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$C^prime subsetneq C$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the same lattice width. Such bodies will be called ‘lattice reduced’, and they occur naturally in the study of the flatness constant in integer programming, as well as other problems related to lattice width. We show that any simplex that realizes the flatness constant must be lattice reduced and prove structural properties of general lattice reduced convex bodies: they are polytopes with at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$2^{d+1}-2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; vertices and their lattice width is attained by at least &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;log&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Omega (log d)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; independent directions. Strongly related to lattice reduced bodies are the ‘lattice complete bodies’, which are convex bodies &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;annotation&gt;$C$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for which there exists no &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;⊋&lt;/mo&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$C^prime supsetneq C$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$C^prime$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has the same lattice diameter as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12982","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142244976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local cone multipliers and Cauchy–Szegö projections in bounded symmetric domains","authors":"Fernando Ballesta Yagüe,&nbsp;Gustavo Garrigós","doi":"10.1112/jlms.12986","DOIUrl":"https://doi.org/10.1112/jlms.12986","url":null,"abstract":"<p>We show that the cone multiplier satisfies local <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> bounds only in the trivial range <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>⩽</mo>\u0000 <mi>q</mi>\u0000 <mo>⩽</mo>\u0000 <mn>2</mn>\u0000 <mo>⩽</mo>\u0000 <mi>p</mi>\u0000 <mo>⩽</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$1leqslant qleqslant 2leqslant pleqslant infty$</annotation>\u0000 </semantics></math>. To do so, we suitably adapt to this setting the proof of Fefferman for the ball multiplier. As a consequence we answer negatively a question by Békollé and Bonami, regarding the continuity from <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>q</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$L^prightarrow L^q$</annotation>\u0000 </semantics></math> of the Cauchy–Szegö projections associated with a class of bounded symmetric domains in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>${mathbb {C}}^n$</annotation>\u0000 </semantics></math> with rank <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$rgeqslant 2$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12986","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142174156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the modulus of continuity of solutions to nonlocal parabolic equations","authors":"Naian Liao","doi":"10.1112/jlms.12985","DOIUrl":"https://doi.org/10.1112/jlms.12985","url":null,"abstract":"<p>A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. Hölder modulus of continuity is then deduced under a slightly stronger tail condition. These regularity estimates are demonstrated under the framework of nonlocal <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-Laplacian with measurable kernels.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12985","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142160292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nijenhuis operators with a unity and \u0000 \u0000 F\u0000 $F$\u0000 -manifolds","authors":"Evgenii I. Antonov,&nbsp;Andrey Yu. Konyaev","doi":"10.1112/jlms.12983","DOIUrl":"https://doi.org/10.1112/jlms.12983","url":null,"abstract":"<p>The core object of this paper is a pair <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>L</mi>\u0000 <mo>,</mo>\u0000 <mi>e</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(L, e)$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> is a Nijenhuis operator and <span></span><math>\u0000 <semantics>\u0000 <mi>e</mi>\u0000 <annotation>$e$</annotation>\u0000 </semantics></math> is a vector field satisfying a specific Lie derivative condition, that is, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>e</mi>\u0000 </msub>\u0000 <mi>L</mi>\u0000 <mo>=</mo>\u0000 <mo>Id</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {L}_{e}L=operatorname{Id}$</annotation>\u0000 </semantics></math>. Our research unfolds in two parts. In the first part, we establish a splitting theorem for Nijenhuis operators with a unity, offering an effective reduction of their study to cases where <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for <span></span><math>\u0000 <semantics>\u0000 <mi>gl</mi>\u0000 <annotation>$mathrm{gl}$</annotation>\u0000 </semantics></math>-regular Nijenhuis operators with a unity around algebraically generic points, along with seminormal forms for dimensions 2 and 3. In the second part, we establish the relationship between Nijenhuis operators with a unity and <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math>-manifolds. Specifically, we prove that the class of regular <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math>-manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math>-manifolds around singularities.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12983","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Valuative invariants for large classes of matroids","authors":"Luis Ferroni,&nbsp;Benjamin Schröter","doi":"10.1112/jlms.12984","DOIUrl":"https://doi.org/10.1112/jlms.12984","url":null,"abstract":"<p>We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a <i>stressed subset</i>. This framework provides a new combinatorial characterization of the class of (elementary) split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which, in turn, can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on these matroids depend solely on the behavior of the invariant on a tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan–Lusztig polynomials, the Whitney numbers of the first and second kinds, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's <span></span><math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$g$</annotation>\u0000 </semantics></math>-polynomials, as well as Chow rings of matroids and their Hilbert–Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12984","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142152281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the intersection form of fillings","authors":"Zhengyi Zhou","doi":"10.1112/jlms.12981","DOIUrl":"https://doi.org/10.1112/jlms.12981","url":null,"abstract":"<p>We prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC in [Lazarev, Geom. Funct. Anal. <b>30</b> (2020), no. 1, 188–254]) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142152222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles","authors":"Stephen Lester,&nbsp;Igor Wigman","doi":"10.1112/jlms.12977","DOIUrl":"https://doi.org/10.1112/jlms.12977","url":null,"abstract":"<p>Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-<span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> disc by its area is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>+</mo>\u0000 <mi>o</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O(R^{1/2+o(1)})$</annotation>\u0000 </semantics></math>. One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12977","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142123045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}