Bulletin of the London Mathematical Society最新文献

{"title":"Smooth structures on nonorientable 4-manifolds via twisting operations","authors":"Valentina Bais, Rafael Torres","doi":"10.1112/blms.70060","DOIUrl":"10.1112/blms.70060","url":null,"abstract":"<p>Five observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> inside <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {R}P^2times S^2$</annotation>\u0000 </semantics></math> such that performing a Gluck twist on <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> produces a manifold <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> that is homeomorphic but not diffeomorphic to the total space of the nontrivial 2-sphere bundle over the real projective plane <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mi>γ</mi>\u0000 <mi>⊕</mi>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$S(2gamma oplus mathbb {R})$</annotation>\u0000 </semantics></math>. The second observation is that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> and a mapping torus that was used by Cappell–Shaneson to construct an exotic <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {R}P^4$</annotation>\u0000 </semantics></math>. This construction of <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> is similar to the one of the Cappell–Shaneson homotopy 4-spheres. The third observation is that twisting an embedded real projective plane inside <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> produces a manifold that is homeomorphic but not diffeomorphic to the circle su","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1768-1790"},"PeriodicalIF":0.9,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244422","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":"Direct and inverse spectral problems for the Schrödinger operator with double generalized Regge boundary conditions","authors":"Xiao-Chuan Xu,&nbsp;Yu-Ting Huang","doi":"10.1112/blms.70054","DOIUrl":"10.1112/blms.70054","url":null,"abstract":"<p>In this paper, we study the direct and inverse spectral problems for the Schrödinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic distribution of the eigenvalues. For the inverse problems, we prove several uniqueness theorems, including the cases: even potential, two-spectra, as well as the general partial inverse problem.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1671-1690"},"PeriodicalIF":0.9,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244423","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":"Bloch functions with wild boundary behavior in \u0000 \u0000 \u0000 C\u0000 N\u0000 \u0000 ${mathbb {C}}^N$","authors":"Stéphane Charpentier,&nbsp;Nicolas Espoullier,&nbsp;Rachid Zarouf","doi":"10.1112/blms.70055","DOIUrl":"10.1112/blms.70055","url":null,"abstract":"<p>We prove the existence of functions <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> in the Bloch space of the unit ball <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>N</mi>\u0000 </msub>\u0000 <annotation>${mathbb {B}}_N$</annotation>\u0000 </semantics></math> of <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 the property that, given any measurable function <span></span><math>\u0000 <semantics>\u0000 <mi>φ</mi>\u0000 <annotation>$varphi$</annotation>\u0000 </semantics></math> on the unit sphere <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>N</mi>\u0000 </msub>\u0000 <annotation>${mathbb {S}}_N$</annotation>\u0000 </semantics></math>, there exists a sequence <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>r</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$(r_n)_n$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>r</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$r_nin (0,1)$</annotation>\u0000 </semantics></math>, converging to 1, such that for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>w</mi>\u0000 <mo>∈</mo>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>N</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$win {mathbb {B}}_N$</annotation>\u0000 </semantics></math>,\u0000\u0000 </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1691-1707"},"PeriodicalIF":0.9,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244427","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 the Zoll deformations of the Kepler problem","authors":"Luca Asselle,&nbsp;Stefano Baranzini","doi":"10.1112/blms.70059","DOIUrl":"10.1112/blms.70059","url":null,"abstract":"<p>A celebrated result of Bertrand states that the only central force potentials on the plane with the property that all bounded orbits are periodic are the Kepler potential and the potential of the harmonic oscillator. In this paper, we complement Bertrand's theorem showing the existence of an infinite-dimensional space of central force potentials on the plane which are <i>Zoll</i> at a given energy level, meaning that all non-collision orbits with given energy are closed and of the same length. We also determine all natural systems on the (not necessarily flat) plane which are invariant under rotations and Zoll at a given energy and prove several existence and rigidity results for systems which are Zoll at multiple energies.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1749-1767"},"PeriodicalIF":0.9,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70059","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244421","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 the double tangent of projective closed curves","authors":"Thomas Blomme","doi":"10.1112/blms.70061","DOIUrl":"10.1112/blms.70061","url":null,"abstract":"<p>We generalize a previous result by Fabricius–Bjerre [Math. Scand. 11 (1962), no. 2, 113–116] from curves 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> to curves in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {R}P^2$</annotation>\u0000 </semantics></math>. Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson–Vogt [Res. Math. Sci. 8 (2021), no. 26] and proves its positivity, conjectured by Larson–Vogt. Our method is not specific to quartics and applies to algebraic curves of any degree.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1791-1800"},"PeriodicalIF":0.9,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244430","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 the class of NY compact spaces of finitely supported elements and related classes","authors":"Antonio Avilés,&nbsp;Mikołaj Krupski","doi":"10.1112/blms.70058","DOIUrl":"10.1112/blms.70058","url":null,"abstract":"&lt;p&gt;We prove that a compact space &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; embeds into a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;σ&lt;/mi&gt;\u0000 &lt;annotation&gt;$sigma$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-product of compact metrizable spaces (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;σ&lt;/mi&gt;\u0000 &lt;annotation&gt;$sigma$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-product of intervals) if and only if &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; is (strongly countable-dimensional) hereditarily metalindelöf and every subspace 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; has a nonempty relative open second countable subset. This provides novel characterizations of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;annotation&gt;$omega$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-Corson and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$NY$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;σ&lt;/mi&gt;\u0000 &lt;annotation&gt;$sigma$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space &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;, the property of being &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$NY$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; compact is determined by the topological structure of the space &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;C&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$C_p(K)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of continuous real-valued functions 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; equipped with the pointwise convergence topology. This refines a r","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1729-1748"},"PeriodicalIF":0.9,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244420","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 completely multiplicative \u0000 \u0000 \u0000 ±\u0000 1\u0000 \u0000 $pm 1$\u0000 sequences that omit many consecutive \u0000 \u0000 \u0000 +\u0000 1\u0000 \u0000 $+1$\u0000 values","authors":"Yichen You","doi":"10.1112/blms.70056","DOIUrl":"10.1112/blms.70056","url":null,"abstract":"<p>We say that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>±</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$pm 1$</annotation>\u0000 </semantics></math>-valued completely multiplicative functions are <i>length-</i><span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> <i>functions</i> <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> if they take the value <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$+1$</annotation>\u0000 </semantics></math> at at most <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> consecutive integers. We introduce a method to extend the length of <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> using the idea of the “rotation trick” in [7]. Under the assumption of Elliott's conjecture, this method allows us to construct length-<span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> functions systematically for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>⩾</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation>$kgeqslant 4$</annotation>\u0000 </semantics></math> which generalizes the work of Schur for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$k = 2$</annotation>\u0000 </semantics></math> and Hudson for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>=</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$k =3$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1708-1717"},"PeriodicalIF":0.9,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70056","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144245148","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 dimensional worm domains","authors":"Simone Calamai,&nbsp;Gian Maria Dall'Ara","doi":"10.1112/blms.70057","DOIUrl":"10.1112/blms.70057","url":null,"abstract":"<p>We show how to construct a class of smooth bounded pseudoconvex domains whose boundary contains a given Stein manifold with strongly pseudoconvex boundary, having a prescribed codimension and D'Angelo class (a cohomological invariant measuring the “winding” of the boundary of the domain around the submanifold). Some open questions in the regularity theory of the <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>∂</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <annotation>$overline{partial }$</annotation>\u0000 </semantics></math>-Neumann problem are discussed in the setting of these domains.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1718-1728"},"PeriodicalIF":0.9,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144245037","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":"Addendum to Herglotz's representation and Carathéodory's approximation","authors":"Tirthankar Bhattacharyya,&nbsp;Mainak Bhowmik,&nbsp;Poornendu Kumar","doi":"10.1112/blms.70053","DOIUrl":"10.1112/blms.70053","url":null,"abstract":"<p>This addendum is to clarify that Theorem 3.1 in <i>Herglotz's representation and Carathéodory's approximation</i> published in Bull. Lond. Math. Soc. 56 (2024), no. 12, 3752–3776 was known.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 5","pages":""},"PeriodicalIF":0.9,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70053","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143938799","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 Lev's periodicity conjecture","authors":"Christian Reiher","doi":"10.1112/blms.70043","DOIUrl":"10.1112/blms.70043","url":null,"abstract":"&lt;p&gt;We classify the sum-free subsets of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;annotation&gt;${mathbb {F}}_3^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; whose density exceeds &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;annotation&gt;$frac{1}{6}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset &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;msubsup&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;${Asubseteq {mathbb {F}}_3^n}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;annotation&gt;$v$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfying &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;v&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$A+v=A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;), then &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&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;1&lt;/mn&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$|A|leqslant frac{1}{2}(3^{n-1}+1)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;—a bound known to be optimal if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;≠&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 5","pages":"1496-1511"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143939665","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}