Pub Date : 2026-04-01Epub Date: 2026-02-10DOI: 10.1016/j.aim.2026.110830
Quan Situ
The hybrid quantum group was firstly introduced by Gaitsgory, whose category can be viewed as a quantum analogue of BGG category . We give a coherent model for its principal block at roots of unity, using the non-commutative Springer resolution defined by Bezrukavnikov–Mirković. In particular, the principal block is derived equivalent to the affine Hecke category. As an application, we endow the principal block with a canonical grading, and show that the graded multiplicity of simple module in Verma module is given by the generic Kazhdan–Lusztig polynomial.
{"title":"Category O for hybrid quantum groups and non-commutative Springer resolutions","authors":"Quan Situ","doi":"10.1016/j.aim.2026.110830","DOIUrl":"10.1016/j.aim.2026.110830","url":null,"abstract":"<div><div>The hybrid quantum group was firstly introduced by Gaitsgory, whose category <span><math><mi>O</mi></math></span> can be viewed as a quantum analogue of BGG category <span><math><mi>O</mi></math></span>. We give a coherent model for its principal block at roots of unity, using the non-commutative Springer resolution defined by Bezrukavnikov–Mirković. In particular, the principal block is derived equivalent to the affine Hecke category. As an application, we endow the principal block with a canonical grading, and show that the graded multiplicity of simple module in Verma module is given by the generic Kazhdan–Lusztig polynomial.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110830"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174567","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}
Pub Date : 2026-04-01Epub Date: 2026-02-12DOI: 10.1016/j.aim.2026.110846
Jian Ding , Ewain Gwynne , Zijie Zhuang
We prove that the set of thick points of the log-correlated Gaussian field contains an unbounded path in sufficiently high dimensions. This contrasts with the two-dimensional case, where Aru, Papon, and Powell (2023) showed that the set of thick points is totally disconnected. This result has an interesting implication for the exponential metric of the log-correlated Gaussian field: in sufficiently high dimensions, when the parameter ξ is large, the set-to-set distance exponent (if it exists) is negative. This suggests that a new phase may emerge for the exponential metric, which does not appear in two dimensions. In addition, we establish similar results for the set of thick points of the branching random walk. As an intermediate result, we also prove that the critical probability for fractal percolation converges to 0 as .
{"title":"Percolation of thick points of the log-correlated Gaussian field in high dimensions","authors":"Jian Ding , Ewain Gwynne , Zijie Zhuang","doi":"10.1016/j.aim.2026.110846","DOIUrl":"10.1016/j.aim.2026.110846","url":null,"abstract":"<div><div>We prove that the set of thick points of the log-correlated Gaussian field contains an unbounded path in sufficiently high dimensions. This contrasts with the two-dimensional case, where Aru, Papon, and Powell (2023) showed that the set of thick points is totally disconnected. This result has an interesting implication for the exponential metric of the log-correlated Gaussian field: in sufficiently high dimensions, when the parameter <em>ξ</em> is large, the set-to-set distance exponent (if it exists) is negative. This suggests that a new phase may emerge for the exponential metric, which does not appear in two dimensions. In addition, we establish similar results for the set of thick points of the branching random walk. As an intermediate result, we also prove that the critical probability for fractal percolation converges to 0 as <span><math><mi>d</mi><mo>→</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110846"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174560","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}
Pub Date : 2026-04-01Epub Date: 2026-02-12DOI: 10.1016/j.aim.2026.110849
Shreyasi Datta , Liyang Shao
We prove the hyperplane absolute winning property of weighted inhomogeneous badly approximable vectors in . This answers a question by Beresnevich–Nesharim–Yang and extends the main result of Beresnevich et al. (2021) [12] to the inhomogeneous set-up.
We also show for any nondegenerate curve and nondegenerate analytic manifold that almost every point is not weighted inhomogeneous badly approximable for any weight. This is achieved by duality and the quantitative nondivergence estimates from homogeneous dynamics motivated by Beresnevich and Yang (2023) [18], together with the methods from arXiv:2307.10109.
{"title":"Winning and nullity of inhomogeneous bad","authors":"Shreyasi Datta , Liyang Shao","doi":"10.1016/j.aim.2026.110849","DOIUrl":"10.1016/j.aim.2026.110849","url":null,"abstract":"<div><div>We prove the hyperplane absolute winning property of weighted inhomogeneous badly approximable vectors in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. This answers a question by Beresnevich–Nesharim–Yang and extends the main result of Beresnevich et al. (2021) <span><span>[12]</span></span> to the inhomogeneous set-up.</div><div>We also show for any nondegenerate curve and nondegenerate analytic manifold that almost every point is not weighted inhomogeneous badly approximable for any weight. This is achieved by duality and the quantitative nondivergence estimates from homogeneous dynamics motivated by Beresnevich and Yang (2023) <span><span>[18]</span></span>, together with the methods from <span><span>arXiv:2307.10109</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110849"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174620","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}
Pub Date : 2026-04-01Epub Date: 2026-02-04DOI: 10.1016/j.aim.2026.110819
Christopher J. Bishop , David L. Bishop
We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval can be uniformly approximated by a real-valued polynomial with only real critical points and whose derivatives converge to zero almost everywhere on I. Alternatively, the approximants may be chosen so that the derivatives converge to plus infinity almost everywhere, or so that these behaviors each occur almost everywhere on specified sets. This extends work by the second author, showing that the derivatives can also be taken to diverge pointwise almost everywhere. Together, these results prove that a 1994 theorem of Clunie and Kuijlaars is sharp.
{"title":"Approximation by singular polynomial sequences","authors":"Christopher J. Bishop , David L. Bishop","doi":"10.1016/j.aim.2026.110819","DOIUrl":"10.1016/j.aim.2026.110819","url":null,"abstract":"<div><div>We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval <span><math><mi>I</mi><mo>⊂</mo><mi>R</mi></math></span> can be uniformly approximated by a real-valued polynomial with only real critical points and whose derivatives converge to zero almost everywhere on <em>I</em>. Alternatively, the approximants may be chosen so that the derivatives converge to plus infinity almost everywhere, or so that these behaviors each occur almost everywhere on specified sets. This extends work by the second author, showing that the derivatives can also be taken to diverge pointwise almost everywhere. Together, these results prove that a 1994 theorem of Clunie and Kuijlaars is sharp.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110819"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174622","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}
Pub Date : 2026-04-01Epub Date: 2026-01-30DOI: 10.1016/j.aim.2026.110832
Rui Han , Wilhelm Schlag
We prove non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori is assumed to be a Diophantine rotation. Results on arithmetic localization are obtained for , and applications to skew shifts, stacked graphene, XY spin chains, and coupled Harper models are presented.
{"title":"Non-perturbative localization for quasi-periodic Jacobi block matrices","authors":"Rui Han , Wilhelm Schlag","doi":"10.1016/j.aim.2026.110832","DOIUrl":"10.1016/j.aim.2026.110832","url":null,"abstract":"<div><div>We prove non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msup></math></span> is assumed to be a Diophantine rotation. Results on arithmetic localization are obtained for <span><math><mi>b</mi><mo>=</mo><mn>1</mn></math></span>, and applications to skew shifts, stacked graphene, XY spin chains, and coupled Harper models are presented.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110832"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081817","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}
Pub Date : 2026-04-01Epub Date: 2026-02-10DOI: 10.1016/j.aim.2026.110844
Inkang Kim , Pierre Pansu , Xueyuan Wan
We show that every integer in the interval is achieved by the signature of a rank 2p flat symplectic bundle over a surface with boundary Σ. When , one can prescribe the type (elliptic, parabolic, hyperbolic) of the holonomy along the boundary.
{"title":"On possible values of the signature of flat symplectic bundles over surfaces with boundary","authors":"Inkang Kim , Pierre Pansu , Xueyuan Wan","doi":"10.1016/j.aim.2026.110844","DOIUrl":"10.1016/j.aim.2026.110844","url":null,"abstract":"<div><div>We show that every integer in the interval <span><math><mo>[</mo><mn>2</mn><mi>p</mi><mi>χ</mi><mo>(</mo><mi>Σ</mi><mo>)</mo><mo>,</mo><mo>−</mo><mn>2</mn><mi>p</mi><mi>χ</mi><mo>(</mo><mi>Σ</mi><mo>)</mo><mo>]</mo></math></span> is achieved by the signature of a rank 2<em>p</em> flat symplectic bundle over a surface with boundary Σ. When <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span>, one can prescribe the type (elliptic, parabolic, hyperbolic) of the holonomy along the boundary.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110844"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146170468","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}
Pub Date : 2026-04-01Epub Date: 2026-01-30DOI: 10.1016/j.aim.2026.110827
Andrea Colesanti , Lei Qin , Paolo Salani
We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted p-operator where , in the class of bounded Lipschitz domains in . We also prove that the corresponding positive eigenfunctions are log-concave if the domain is convex.
{"title":"Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities","authors":"Andrea Colesanti , Lei Qin , Paolo Salani","doi":"10.1016/j.aim.2026.110827","DOIUrl":"10.1016/j.aim.2026.110827","url":null,"abstract":"<div><div>We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted <em>p</em>-operator<span><span><span><math><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>γ</mi></mrow></msub><mi>u</mi><mo>=</mo><mo>−</mo><mtext>div</mtext><mo>(</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>∇</mi><mi>u</mi><mo>)</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span>, in the class of bounded Lipschitz domains in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We also prove that the corresponding positive eigenfunctions are log-concave if the domain is convex.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110827"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080761","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}
Pub Date : 2026-04-01Epub Date: 2026-01-28DOI: 10.1016/j.aim.2026.110820
Noah Kravitz , Borys Kuca , James Leng
Using PET and quantitative concatenation techniques, we establish box-norm control with the “expected” directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper [20], we complete this program for sets in lacking nondegenerate progressions of the form , where is any fixed polynomial with an integer root of multiplicity 1.
{"title":"Quantitative concatenation for polynomial box norms","authors":"Noah Kravitz , Borys Kuca , James Leng","doi":"10.1016/j.aim.2026.110820","DOIUrl":"10.1016/j.aim.2026.110820","url":null,"abstract":"<div><div>Using PET and quantitative concatenation techniques, we establish box-norm control with the “expected” directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper <span><span>[20]</span></span>, we complete this program for sets in <span><math><msup><mrow><mo>[</mo><mi>N</mi><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> lacking nondegenerate progressions of the form <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>x</mi><mo>+</mo><mi>P</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>y</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>+</mo><mi>P</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>P</mi><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>z</mi><mo>]</mo></math></span> is any fixed polynomial with an integer root of multiplicity 1.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110820"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080762","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}
Pub Date : 2026-04-01Epub Date: 2026-01-30DOI: 10.1016/j.aim.2026.110829
Daniel McGinnis , Nikola Sadovek
We solve a long-standing open problem posed by Goodman & Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in to admit a k-transversal for any . This result is a common generalization of Helly's theorem () and the Goodman-Pollack-Wenger theorem (). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex k-transversal to a family of convex sets in , extending the work of McGinnis (). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.
{"title":"A necessary and sufficient condition for k-transversals","authors":"Daniel McGinnis , Nikola Sadovek","doi":"10.1016/j.aim.2026.110829","DOIUrl":"10.1016/j.aim.2026.110829","url":null,"abstract":"<div><div>We solve a long-standing open problem posed by Goodman & Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> to admit a <em>k</em>-transversal for any <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>d</mi><mo>−</mo><mn>1</mn></math></span>. This result is a common generalization of Helly's theorem (<span><math><mi>k</mi><mo>=</mo><mn>0</mn></math></span>) and the Goodman-Pollack-Wenger theorem (<span><math><mi>k</mi><mo>=</mo><mi>d</mi><mo>−</mo><mn>1</mn></math></span>). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex <em>k</em>-transversal to a family of convex sets in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, extending the work of McGinnis (<span><math><mi>k</mi><mo>=</mo><mi>d</mi><mo>−</mo><mn>1</mn></math></span>). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110829"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081946","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}
Pub Date : 2026-04-01Epub Date: 2026-01-22DOI: 10.1016/j.aim.2026.110788
Ulrich Bunke , Daniel Kasprowski , Christoph Winges
We prove the Farrell–Jones conjecture for finitary localising invariants with coefficients in left-exact ∞-categories for finitely -amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over -ring spectra.
{"title":"On the Farrell–Jones conjecture for localising invariants","authors":"Ulrich Bunke , Daniel Kasprowski , Christoph Winges","doi":"10.1016/j.aim.2026.110788","DOIUrl":"10.1016/j.aim.2026.110788","url":null,"abstract":"<div><div>We prove the Farrell–Jones conjecture for finitary localising invariants with coefficients in left-exact ∞-categories for finitely <span><math><mi>F</mi></math></span>-amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-ring spectra.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110788"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146006653","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}
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