Advances in Mathematics最新文献

{"title":"Universality of L-functions over function fields","authors":"Julio C. Andrade ,&nbsp;Steven M. Gonek ,&nbsp;Yoonbok Lee","doi":"10.1016/j.aim.2025.110265","DOIUrl":"10.1016/j.aim.2025.110265","url":null,"abstract":"<div><div>We prove that Dirichlet <em>L</em>-functions corresponding to Dirichlet characters for <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> with <em>q</em> odd are universal in the following sense. Let <span><math><mi>Q</mi></math></span> denote either the set of all prime polynomials <em>Q</em> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, or the set of all polynomials <em>Q</em> that are products of a fixed set of prime polynomials <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. Let <em>U</em> be the open rectangle with vertices <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>i</mi><mi>α</mi><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>i</mi><mi>α</mi><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>i</mi><mi>β</mi><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>i</mi><mi>β</mi></math></span>, where <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&lt;</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mn>1</mn></math></span> and <span><math><mn>0</mn><mo>&lt;</mo><mi>β</mi><mo>−</mo><mi>α</mi><mo>≤</mo><mn>2</mn><mi>π</mi><mo>/</mo><mo>(</mo><mn>3</mn><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>)</mo></math></span>. Suppose also that <em>C</em> is a compact set in <em>U</em> with positive Lebesgue measure whose complement is connected and that <em>f</em> is a prescribed continuous, nonvanishing function on <em>C</em> that is analytic on the interior of <em>C</em>. Then if <span><math><mi>Q</mi><mo>∈</mo><mi>Q</mi></math></span> is of high enough degree, a positive proportion of the <em>L</em>-functions with characters to this modulus approximate <em>f</em> arbitrarily closely. This extends for the first time (as far as we know) the notion of universality of <em>L</em>-functions over number fields to the function field setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110265"},"PeriodicalIF":1.5,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143828307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Categorical action for finite classical groups and its applications: Characteristic 0","authors":"Pengcheng Li ,&nbsp;Peng Shan ,&nbsp;Jiping Zhang","doi":"10.1016/j.aim.2025.110275","DOIUrl":"10.1016/j.aim.2025.110275","url":null,"abstract":"<div><div>In this paper, we construct a categorical double quantum Heisenberg action on the representation category of finite classical groups <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>O</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mo>±</mo></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo></math></span> with <em>q</em> odd. Over a field of characteristic zero or characteristic <em>ℓ</em> with <span><math><mi>ℓ</mi><mo>∤</mo><mi>q</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, we deduce a categorical action of a Kac-Moody algebra <span><math><mi>s</mi><msubsup><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>I</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow><mrow><mo>′</mo></mrow></msubsup><mo>⊕</mo><mi>s</mi><msubsup><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>I</mi></mrow><mrow><mo>−</mo></mrow></msub></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> on the representation category of finite classical groups. We show that the colored weight functions <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>u</mi><mo>)</mo><mo>(</mo><mo>−</mo><mo>)</mo></math></span>, <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>(</mo><mi>v</mi><mo>)</mo><mo>(</mo><mo>−</mo><mo>)</mo></math></span> and uniform projection can distinguish all irreducible characters of finite classical groups. In particular, the colored weight functions are complete invariants of quadratic unipotent characters. We also show that using the theta correspondence and extra symmetries of categorical double quantum Heisenberg action, the Kac-Moody action on the Grothendieck group of the whole category can be determined explicitly.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110275"},"PeriodicalIF":1.5,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Classification of global solutions to the obstacle problem in the plane","authors":"Anthony Salib ,&nbsp;Georg S. Weiss","doi":"10.1016/j.aim.2025.110276","DOIUrl":"10.1016/j.aim.2025.110276","url":null,"abstract":"<div><div>Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110276"},"PeriodicalIF":1.5,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A unified approach to mass transference principle and large intersection property","authors":"Yubin He","doi":"10.1016/j.aim.2025.110267","DOIUrl":"10.1016/j.aim.2025.110267","url":null,"abstract":"<div><div>The mass transference principle, discovered by Beresnevich and Velani (2006) <span><span>[5]</span></span>, is a landmark result in metric Diophantine approximation that allows us to obtain the Hausdorff measure theory of <span><math><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>sup</mi></mrow></mrow></math></span> sets. Another important tool is the notion of a large intersection property, introduced and systematically studied by Falconer (1994) <span><span>[9]</span></span>. The former mainly focuses on passing between full (Lebesgue) measure and full Hausdorff measure statements, while the latter transfers full Hausdorff content statement to Hausdorff dimension. From this perspective, the proofs of the two results are similar but often treated in different ways.</div><div>In this paper, we establish a general mass transference principle from the viewpoint of Hausdorff content, aiming to provide a unified proof for the aforementioned results. More precisely, this principle enables us to transfer the Hausdorff content bounds of a sequence of open sets <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to the full Hausdorff measure statement and the large intersection property for <span><math><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>sup</mi></mrow></mrow><mspace></mspace><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. One of the advantages of our approach is that the verification of the Hausdorff content bound does not require the construction of Cantor-like subset, resulting in a much simpler proof. As an application, we provide simpler proofs for several mass transference principles.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110267"},"PeriodicalIF":1.5,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143821396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Cauchy problem for 3D Navier-Stokes helical vortex filament","authors":"Francisco Gancedo,&nbsp;Antonio Hidalgo-Torné","doi":"10.1016/j.aim.2025.110268","DOIUrl":"10.1016/j.aim.2025.110268","url":null,"abstract":"<div><div>This paper studies the Cauchy problem for a helical vortex filament evolving by the 3D incompressible Navier-Stokes equations. We prove global-in-time well-posedness and smoothing of solutions with initial vorticity concentrated on a helix. We provide a local-in-time well-posedness result for vortex filaments periodic in one spatial direction, and show that solutions with helical initial data preserve this symmetry. We follow the approach of <span><span>[4]</span></span>, where the analogue local-in-time result has been obtained for closed vortex filaments in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Next, we apply local energy weak solutions theory with a novel estimate for helical functions in non-helical domains to uniquely extend the solutions globally in time. This is the first global-in-time well-posedness result for a vortex filament without size restriction and without vanishing swirl assumptions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110268"},"PeriodicalIF":1.5,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143816931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The global Σn+21-uniformization property and BPFA","authors":"Stefan Hoffelner","doi":"10.1016/j.aim.2025.110272","DOIUrl":"10.1016/j.aim.2025.110272","url":null,"abstract":"<div><div>We show that given a reflecting cardinal, one can produce a model of <span><math><mi>BPFA</mi></math></span> where the <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-uniformization property holds simultaneously for all <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> which stands in stark constrast to the situation under <span><math><mi>PFA</mi></math></span>. We do this via a new forcing construction which gets rid of the delimitations caused by a good projective well-order of the reals, which has been the sole tool to obtain global Σ-uniformization since 1959.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"470 ","pages":"Article 110272"},"PeriodicalIF":1.5,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Roots of Alexander polynomials of random positive 3-braids","authors":"Nathan M. Dunfield ,&nbsp;Giulio Tiozzo","doi":"10.1016/j.aim.2025.110254","DOIUrl":"10.1016/j.aim.2025.110254","url":null,"abstract":"<div><div>Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110254"},"PeriodicalIF":1.5,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hausdorff dimensions of affine multiplicative shifts","authors":"Jung-Chao Ban ,&nbsp;Wen-Guei Hu ,&nbsp;Guan-Yu Lai ,&nbsp;Lingmin Liao","doi":"10.1016/j.aim.2025.110266","DOIUrl":"10.1016/j.aim.2025.110266","url":null,"abstract":"<div><div>We calculate the Minkowski and Hausdorff dimensions of affine multiplicative shifts<span><span><span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mi>A</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>;</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></msubsup><mo>=</mo><mo>{</mo><msubsup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>∈</mo><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>N</mi></mrow></msup><mspace></mspace><mspace></mspace><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>p</mi><mi>k</mi><mo>+</mo><mi>a</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mi>k</mi><mo>+</mo><mi>b</mi></mrow></msub></mrow></msub><mo>=</mo><mn>1</mn><mtext> for all </mtext><mi>k</mi><mo>≥</mo><mn>1</mn><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>&lt;</mo><mi>q</mi><mo>∈</mo><mi>N</mi></math></span>, <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>Z</mi></math></span>, <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>+</mo><mi>a</mi><mo>&lt;</mo><mi>q</mi><mo>+</mo><mi>b</mi></math></span> and <span><math><mi>A</mi><mo>=</mo><msubsup><mrow><mo>[</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>]</mo></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup></math></span> is an <span><math><mi>m</mi><mo>×</mo><mi>m</mi></math></span> matrix with entries 0 or 1.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110266"},"PeriodicalIF":1.5,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Maxwell-Klein-Gordon equation with scattering data","authors":"Wei Dai ,&nbsp;He Mei ,&nbsp;Dongyi Wei ,&nbsp;Shiwu Yang","doi":"10.1016/j.aim.2025.110271","DOIUrl":"10.1016/j.aim.2025.110271","url":null,"abstract":"<div><div>It has been shown in <span><span>[59]</span></span> that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>3</mn></mrow></msup></math></span> decay like linear solutions. One hence can define the associated radiation field on the future null infinity as the limit of <span><math><mo>(</mo><mi>r</mi><munder><mrow><mi>α</mi></mrow><mo>_</mo></munder><mo>,</mo><mi>r</mi><mi>ϕ</mi><mo>)</mo></math></span> along the out going null geodesics. In this paper, we show the existence of a global solution to the MKG system which scatters to any given sufficiently localized radiation field with arbitrarily large size and total charge. The result follows by studying the characteristic initial value problem for the MKG system with general large data by using gauge invariant vector field method. We in particular extend the small data result of He in <span><span>[35]</span></span> to a class of general large data.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110271"},"PeriodicalIF":1.5,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quivers and the Adams spectral sequence","authors":"Robert Burklund ,&nbsp;Piotr Pstrągowski","doi":"10.1016/j.aim.2025.110270","DOIUrl":"10.1016/j.aim.2025.110270","url":null,"abstract":"<div><div>In this paper, we describe a novel way of identifying Adams spectral sequence <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-terms in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on descent-flatness, bearing on a varied array of ring spectra. In the particular case of <em>p</em>-local integral homology, we are able to give a decomposition of the <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-term, describing it completely in terms of the classical Adams spectral sequence. In the appendix, which can be read independently from the main body of the text, we develop functoriality of deformations of ∞-categories of the second author and Patchkoria.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110270"},"PeriodicalIF":1.5,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}