Bulletin of the London Mathematical Society最新文献

{"title":"Optimal power-weighted Birman–Hardy–Rellich-type inequalities on finite intervals and annuli","authors":"Fritz Gesztesy,&nbsp;Michael M. H. Pang","doi":"10.1112/blms.70063","DOIUrl":"10.1112/blms.70063","url":null,"abstract":"<p>We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power-weighted Birman–Hardy–Rellich-type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one-dimensional Hardy-type result in differential form to derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman–Hardy–Rellich-type sequence of power-weighted higher order Hardy-type inequalities for annuli. In the limit as the annulus approaches <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>${mathbb {R}}^n$</annotation>\u0000 </semantics></math>{0}, we recover well-known prior results on Rellich-type inequalities on <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>${mathbb {R}}^n$</annotation>\u0000 </semantics></math>{0}.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1819-1840"},"PeriodicalIF":0.9,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244989","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":"Eigenvalues of matrix products","authors":"Richard Kenyon,&nbsp;Nicholas Ovenhouse","doi":"10.1112/blms.70071","DOIUrl":"10.1112/blms.70071","url":null,"abstract":"&lt;p&gt;We study pairs of matrices &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;B&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;GL&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$A,Bin mathrm{GL}_n({mathbb {C}})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that the eigenvalues of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;annotation&gt;$B$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and of the product &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;mi&gt;B&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$AB$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are specified in advance. We show that the space of such pairs &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;A&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(A,B)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; under simultaneous conjugation has dimension &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;n&lt;/mi&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;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(n-1)(n-2)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and give an explicit parameterization. More generally, let &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; be a surface of genus &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;annotation&gt;$g$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with &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; punctures. We find a parameterization of the space &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;g&lt;/mi&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;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$Omeg","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 7","pages":"1938-1951"},"PeriodicalIF":0.9,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144681406","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":"Removing scalar curvature assumption for Ricci flow smoothing","authors":"Adam Martens","doi":"10.1112/blms.70073","DOIUrl":"10.1112/blms.70073","url":null,"abstract":"<p>In recent work of Chan–Huang–Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov–Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 7","pages":"1968-1989"},"PeriodicalIF":0.9,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70073","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144681405","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":"Preservation for generation along the structure morphism of coherent algebras over a scheme","authors":"Anirban Bhaduri,&nbsp;Souvik Dey,&nbsp;Pat Lank","doi":"10.1112/blms.70066","DOIUrl":"10.1112/blms.70066","url":null,"abstract":"<p>This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. Additionally, we establish that the Krull dimension of a variety over a field is a lower bound for the Rouquier dimension of the bounded derived category associated with a noncommutative coherent algebra on it. This is an extension of a classical result of Rouquier to the noncommutative context.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1885-1896"},"PeriodicalIF":0.9,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70066","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244620","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":"Corrigendum to “On the existence of critical compatible metrics on contact 3-manifolds,”","authors":"Y. Mitsumatsu,&nbsp;D. Peralta-Salas,&nbsp;R. Slobodeanu","doi":"10.1112/blms.70067","DOIUrl":"10.1112/blms.70067","url":null,"abstract":"<p>A gap in the proof of the main result (Theorem 1.3) in our paper [Bull. Lond. Math. Soc. <b>57</b> (2025), 79–95] is identified and fixed. The gap is related to possible non-orientability of the line bundles defined by eigendirections of the (1,1)-tensor <span></span><math>\u0000 <semantics>\u0000 <mi>h</mi>\u0000 <annotation>$h$</annotation>\u0000 </semantics></math>. </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1918-1920"},"PeriodicalIF":0.9,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70067","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244189","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":"Geometric invariant theory for \u0000 \u0000 \u0000 \u0000 G\u0000 a\u0000 \u0000 ⋊\u0000 \u0000 G\u0000 m\u0000 \u0000 \u0000 $mathbb {G}_artimes mathbb {G}_m$","authors":"Yikun Qiao","doi":"10.1112/blms.70065","DOIUrl":"10.1112/blms.70065","url":null,"abstract":"<p>We study geometric invariant theory for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <msub>\u0000 <mo>⋊</mo>\u0000 <mi>d</mi>\u0000 </msub>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$mathbb {G}_artimes _dmathbb {G}_m$</annotation>\u0000 </semantics></math> over characteristic zero. For a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <msub>\u0000 <mo>⋊</mo>\u0000 <mi>d</mi>\u0000 </msub>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$mathbb {G}_artimes _dmathbb {G}_m$</annotation>\u0000 </semantics></math>-action on a variety <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> in suitable case, we provide an equivariant birational modification <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>:</mo>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation>$p:C^{prime }rightarrow C$</annotation>\u0000 </semantics></math> such that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <mo>/</mo>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$C^{prime }rightarrow C^{prime }/mathbb {G}_a$</annotation>\u0000 </semantics></math> is a principal bundle. The situation covers quotients of unstable strata of any linear <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>SL</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$mathrm{SL}_2$</annotation>\u0000 </semantics></math>-action.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1856-1884"},"PeriodicalIF":0.9,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244164","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":"Smallest totient in a residue class","authors":"Abhishek Jha","doi":"10.1112/blms.70069","DOIUrl":"10.1112/blms.70069","url":null,"abstract":"<p>We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>,</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(m,a)$</annotation>\u0000 </semantics></math> such that <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math> is odd, there exists <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩽</mo>\u0000 <msup>\u0000 <mi>m</mi>\u0000 <mrow>\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 </mrow>\u0000 <annotation>$nleqslant m^{2+o(1)}$</annotation>\u0000 </semantics></math> such that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 <mo>≡</mo>\u0000 <mi>a</mi>\u0000 <mspace></mspace>\u0000 <mo>(</mo>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$varphi (n)equiv a (mathrm{mod} m)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1908-1917"},"PeriodicalIF":0.9,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70069","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244210","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":"Flat cotorsion modules are exchange","authors":"Manuel Cortés-Izurdiaga,&nbsp;Pedro A. Guil Asensio,&nbsp;Ashish K. Srivastava","doi":"10.1112/blms.70068","DOIUrl":"10.1112/blms.70068","url":null,"abstract":"<p>It is known that flat cotorsion modules satisfy the finite exchange property. In this paper, we provide a general criterion that, in particular, demonstrates that flat cotorsion modules indeed satisfy the (full) exchange property.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1897-1907"},"PeriodicalIF":0.9,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244510","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":"The small-scale limit of magnitude and the one-point property","authors":"Emily Roff,&nbsp;Masahiko Yoshinaga","doi":"10.1112/blms.70064","DOIUrl":"10.1112/blms.70064","url":null,"abstract":"<p>The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the <i>one-point property</i> if its magnitude converges to 1 as the space is scaled down to a point. Not every finite metric space has the one-point property: to date, exactly one example has been found of a finite space for which the property fails. Understanding the failure of the one-point property is of interest in clarifying the interpretation of magnitude and its stability with respect to the Gromov–Hausdorff topology. We prove that the one-point property holds generically for finite metric spaces, but that when it fails, the failure can be arbitrarily bad: the small-scale limit of magnitude can take arbitrary real values greater than 1.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1841-1855"},"PeriodicalIF":0.9,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70064","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244438","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 an Erdős similarity problem in the large","authors":"Xiang Gao,&nbsp;Yuveshen Mooroogen,&nbsp;Chi Hoi Yip","doi":"10.1112/blms.70062","DOIUrl":"10.1112/blms.70062","url":null,"abstract":"&lt;p&gt;In a recent paper, Kolountzakis and Papageorgiou ask if for every &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&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;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;annotation&gt;$epsilon in (0,1]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, there exists a set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mo&gt;⊆&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$S subseteq mathbb {R}$&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;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mo&gt;∩&lt;/mo&gt;\u0000 &lt;mi&gt;I&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&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;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$vert S cap Ivert geqslant 1 - epsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for every interval &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;I&lt;/mi&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$I subset mathbb {R}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with unit length, but that does not contain any affine copy of a given increasing sequence of exponential growth or faster. This question is an analog of the well-known Erdős similarity problem. In this paper, we show that for each sequence of real numbers whose integer parts form a set of positive upper Banach density, one can explicitly construct such a set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;annotation&gt;$S$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that contains no affine copy of that sequence. Since there exist sequences of arbitrarily rapid growth that satisfy this condition, our result answers Kolountzakis and Papageorgiou's question in the affirmative. A key ingredient of our proof is a generalization of results by Amice, Kahane, and Haight from metric number theory. In addition, we construct a set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;annotation&gt;$S$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with the required property — but with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&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;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 6","pages":"1801-1818"},"PeriodicalIF":0.9,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70062","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144244280","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}