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

{"title":"On the Fourier transform of random Bernoulli convolutions","authors":"Simon Baker,&nbsp;Henna Koivusalo,&nbsp;Sascha Troscheit,&nbsp;Xintian Zhang","doi":"10.1112/jlms.70515","DOIUrl":"https://doi.org/10.1112/jlms.70515","url":null,"abstract":"<p>We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution\u0000\u0000 </p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70515","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147708067","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":"Expansion of normal subsets of odd-order elements in finite groups","authors":"Chris Parker,&nbsp;Jack Saunders","doi":"10.1112/jlms.70534","DOIUrl":"https://doi.org/10.1112/jlms.70534","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> be a finite group and <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math> a normal subset consisting of odd-order elements. The rational closure of <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>, denoted <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <mi>K</mi>\u0000 </msub>\u0000 <annotation>$mathbf {D}_K$</annotation>\u0000 </semantics></math>, is the set of elements <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>∈</mo>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$x in G$</annotation>\u0000 </semantics></math> with the property that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⟨</mo>\u0000 <mi>x</mi>\u0000 <mo>⟩</mo>\u0000 <mo>=</mo>\u0000 <mo>⟨</mo>\u0000 <mi>y</mi>\u0000 <mo>⟩</mo>\u0000 </mrow>\u0000 <annotation>$langle x rangle = langle y rangle$</annotation>\u0000 </semantics></math> for some <span></span><math>\u0000 <semantics>\u0000 <mi>y</mi>\u0000 <annotation>$y$</annotation>\u0000 </semantics></math> in <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>. If <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>K</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>⊆</mo>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <mi>K</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$K^2 subseteq mathbf {D}_K$</annotation>\u0000 </semantics></math>, we prove that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⟨</mo>\u0000 <mi>K</mi>\u0000 <mo>⟩</mo>\u0000 </mrow>\u0000 <annotation>$langle K rangle$</annotation>\u0000 </semantics></math> is soluble.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70534","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707985","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":"Unitarily invariant valuations on convex functions","authors":"Jonas Knoerr","doi":"10.1112/jlms.70533","DOIUrl":"https://doi.org/10.1112/jlms.70533","url":null,"abstract":"<p>Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on <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> that are invariant under the unitary group are investigated. It is shown that elements belonging to the dense subspace of smooth valuations admit a unique integral representation in terms of two families of Monge–Ampère-type operators. In addition, it is proved that homogeneous valuations are uniquely determined by restrictions to subspaces of appropriate dimension and that this information is encoded in the Fourier–Laplace transform of the associated Goodey–Weil distributions. These results are then used to show that a continuous unitarily invariant valuation is uniquely determined by its restriction to a certain finite family of subspaces 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>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70533","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707986","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":"Multiple front and pulse solutions in spatially periodic systems","authors":"Lukas Bengel,&nbsp;Björn de Rijk","doi":"10.1112/jlms.70530","DOIUrl":"https://doi.org/10.1112/jlms.70530","url":null,"abstract":"<p>In this paper, we develop a comprehensive mathematical toolbox for the construction and spectral stability analysis of stationary multiple front and pulse solutions to general semilinear evolution problems on the real line with spatially periodic coefficients. Starting from a collection of <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> nondegenerate primary front solutions with matching periodic end states, we realize multifront solutions near a formal concatenation of these <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> primary fronts, provided the distances between the front interfaces is sufficiently large. Moreover, we prove that nondegenerate primary pulses are accompanied by periodic pulse solutions of large spatial period. We show that spectral (in)stability properties of the underlying primary fronts or pulses are inherited by the bifurcating multifronts or periodic pulse solutions. The existence and spectral analyses rely on contraction-mapping arguments and Evans-function techniques, leveraging exponential dichotomies to characterize invertibility and Fredholm properties. To demonstrate the applicability of our methods, we analyze the existence and stability of multifronts and periodic pulse solutions in some benchmark models, such as the Gross–Pitaevskii equation with periodic potential and a Klausmeier reaction-diffusion-advection system, thereby identifying novel classes of (stable) solutions. In particular, our methods yield the first spectral and orbital stability result of periodic waves in the focusing Gross–Pitaevskii equation with periodic potential, as well as new instability criteria for multipulse solutions to this equation.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70530","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707948","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 amenability constants of Fourier algebras: new bounds and new examples","authors":"Y. Choi,&nbsp;M. Ghandehari","doi":"10.1112/jlms.70518","DOIUrl":"https://doi.org/10.1112/jlms.70518","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> be a locally compact group. If <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is finite, then the amenability constant of its Fourier algebra, denoted by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>AM</mi>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm AM}({rm A}(G))$</annotation>\u0000 </semantics></math>, admits an explicit formula [Johnson, J. Lond. Math. Soc. 1994]; if <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is infinite, then no such formula for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>AM</mi>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm AM}({rm A}(G))$</annotation>\u0000 </semantics></math> is known, although lower and upper bounds were established by Runde [Proc. Am. Math. Soc. 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>AM</mi>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm AM}({rm A}(G))$</annotation>\u0000 </semantics></math> when <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is discrete. Combining this with previous work of the first author [Choi, Int. Math. Res. Not. 2023], we exhibit new examples of discrete groups and compact groups where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>AM</mi>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm AM}({rm A}(G))$</annotation>\u0000 </semantics></math> can be calculated explicitly; previously this was only known for groups that are products of finite groups with “degenerate” cases. Our new examples also provide additional evidence to support the conjecture that Runde's lower bound for the amenability constant is in fact an equality.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707877","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":"Graphical small cancellation and hyperfiniteness of boundary actions","authors":"Chris Karpinski,&nbsp;Damian Osajda,&nbsp;Koichi Oyakawa","doi":"10.1112/jlms.70516","DOIUrl":"https://doi.org/10.1112/jlms.70516","url":null,"abstract":"<p>We study actions of (infinitely presented) graphical small cancellation groups on the Gromov boundaries of their coned-off Cayley graphs. We show that a class of graphical small cancellation groups, including (infinitely presented) classical small cancellation groups, admit hyperfinite boundary actions, more precisely, the orbit equivalence relation that they induce on the boundaries of the coned-off Cayley graphs is hyperfinite.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70516","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707875","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":"A P-adic class formula for Anderson t-modules","authors":"Alexis Lucas","doi":"10.1112/jlms.70529","DOIUrl":"https://doi.org/10.1112/jlms.70529","url":null,"abstract":"&lt;p&gt;In 2012, Taelman proved a class formula for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$L$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-series associated to Drinfeld &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;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;θ&lt;/mi&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbb {F}_q[theta]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-modules and considered it as a function field analogue of the Birch and Swinnerton-Dyer conjecture. Since then, Taelman's class formula has been generalized to the setting of Anderson &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;annotation&gt;$t$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-modules. Let &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;$P$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a monic irreducible polynomial of &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;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;θ&lt;/mi&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbb {F}_q[theta]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we define the &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;$P$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-adic &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$L$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-series associated with Anderson &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;annotation&gt;$t$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-modules and prove a &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;$P$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-adic class formula à la Taelman linking a &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;$P$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-adic regulator, the class module and a local factor at &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;$P$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Next, we study the vanishing of the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70529","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707876","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":"Function spaces for decoupling","authors":"Andrew Hassell,&nbsp;Pierre Portal,&nbsp;Jan Rozendaal,&nbsp;Po-Lam Yung","doi":"10.1112/jlms.70503","DOIUrl":"https://doi.org/10.1112/jlms.70503","url":null,"abstract":"<p>We introduce new function spaces <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>L</mi>\u0000 <mrow>\u0000 <mi>W</mi>\u0000 <mo>,</mo>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {L}_{W,s}^{q,p}(mathbb {R}^{n})$</annotation>\u0000 </semantics></math> that yield a natural reformulation of the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>ℓ</mi>\u0000 <mi>q</mi>\u0000 </msup>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$ell ^{q}L^{p}$</annotation>\u0000 </semantics></math> decoupling inequalities for the sphere and the light cone. These spaces are invariant under the Euclidean half-wave propagators, but not under all Fourier integral operators unless <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation>$p=q$</annotation>\u0000 </semantics></math>, in which case they coincide with the Hardy spaces for Fourier integral operators. We use these spaces to obtain improvements of the classical fractional integration theorem and local smoothing estimates.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70503","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707887","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":"A Miyaoka–Yau inequality for hyperplane arrangements in \u0000 \u0000 \u0000 CP\u0000 n\u0000 \u0000 $mathbb {CP}^n$","authors":"Martin de Borbon,&nbsp;Dmitri Panov","doi":"10.1112/jlms.70525","DOIUrl":"https://doi.org/10.1112/jlms.70525","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {H}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a hyperplane arrangement in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;CP&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {CP}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We define a quadratic form &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; on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {R}^{mathcal {H}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that is entirely determined by the intersection poset of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {H}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Using the Bogomolov–Gieseker inequality for parabolic bundles, we show that if &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;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbf {a}in mathbb {R}^{mathcal {H}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is such that the weighted arrangement &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;H&lt;/mi&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;annotation&gt;$(mathcal {H}, mathbf {a})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is &lt;i&gt;stable&lt;/i&gt;, then &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;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Q(mathbf {a}) leqslant 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. As an application, we consider the symmetric case where all the weights are equal. The inequality &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;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&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;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotat","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70525","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707886","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":"Supersimplicity and arithmetic progressions","authors":"Amador Martin-Pizarro,&nbsp;Daniel Palacín","doi":"10.1112/jlms.70531","DOIUrl":"https://doi.org/10.1112/jlms.70531","url":null,"abstract":"<p>The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corput's theorem on arithmetic progressions of length 3) with well-known tools and theorems for definable groups in simple theories. In the last sections of this article, we apply our model-theoretic results to bound the number of initial points starting few arithmetic progressions of length 3 in the structure of the additive group of integers with a predicate for the prime integers, assuming Dickson's conjecture, or with a predicate for the square-free integers, as well as for asymptotic limits of finite fields. Our techniques yield similar results for the elements appearing as distances in skew-corners and for Sárközy's theorem on the distance of distinct elements being perfect squares.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147707888","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}