Advances in Mathematics最新文献

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The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110062

Eduard Stefanescu

Let

{(a_{n})}_{n \in N}

be a lacunary sequence satisfying the Hadamard gap condition. We give upper bounds for the maximal gap of the set of dilates

{a_{n} α}_{n \leq N}

modulo 1, in terms of N. For any lacunary sequence

{(a_{n})}_{n \in N}

we prove the existence of a dilation factor α such that the maximal gap is of order at most

(\log N) / N

, and we prove that for Lebesgue almost all α the maximal gap is of order at most

{(\log N)}^{2 + ε} / N

. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.

{"title":"The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation","authors":"Eduard Stefanescu","doi":"10.1016/j.aim.2024.110062","DOIUrl":"10.1016/j.aim.2024.110062","url":null,"abstract":"<div><div>Let <math><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math> be a lacunary sequence satisfying the Hadamard gap condition. We give upper bounds for the maximal gap of the set of dilates <math><msub><mrow><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>α</mi><mo>}</mo></mrow><mrow><mi>n</mi><mo>≤</mo><mi>N</mi></mrow></msub></math> modulo 1, in terms of N. For any lacunary sequence <math><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math> we prove the existence of a dilation factor α such that the maximal gap is of order at most <math><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>)</mo><mo>/</mo><mi>N</mi></math>, and we prove that for Lebesgue almost all α the maximal gap is of order at most <math><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>/</mo><mi>N</mi></math>. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110062"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164283","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}

引用次数: 0

Answers to questions of Grünbaum and Loewner

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110081

Sergii Myroshnychenko , Kateryna Tatarko , Vladyslav Yaskin

We construct a convex body K in

R^{n}

n \geq 5

, with the property that there is exactly one hyperplane H passing through

c (K)

, the centroid of K, such that the centroid of

K \cap H

coincides with

c (K)

. This provides answers to questions of Grünbaum and Loewner for

n \geq 5

. The proof is based on the existence of non-intersection bodies in these dimensions.

引用次数: 0

Subcritical epidemics on random graphs

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110102

Oanh Nguyen , Allan Sly

We study the contact process on random graphs with low infection rate λ. For random d-regular graphs, it is known that the survival time is

O (\log n)

below the critical

λ_{c}

. By contrast, on the Erdős-Rényi random graphs

G (n, d / n)

, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time

n^{λ^{2 + o (1)}}

. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a sharper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.

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引用次数: 0

Maps, simple groups, and arc-transitive graphs

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110086

Martin W. Liebeck , Cheryl E. Praeger

We determine all factorisations

X = A B

, where X is a finite almost simple group and

A, B

are core-free subgroups such that

A \cap B

is cyclic or dihedral. As a main application, we classify the graphs Γ admitting an almost simple arc-transitive group X of automorphisms, such that Γ has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup G of X. We prove that apart from the case where X and G have socles

A_{n}

and

A_{n - 1}

respectively, the only such graphs are the complete graphs

K_{n}

with n a prime power, the Johnson graphs

J (n, 2)

with

n - 1

a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings.

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引用次数: 0

Making the motivic group structure on the endomorphisms of the projective line explicit

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110080

Viktor Balch Barth , William Hornslien , Gereon Quick , Glen Matthew Wilson

We construct a group structure on the set of pointed naive homotopy classes of scheme morphisms from the Jouanolou device to the projective line. The group operation is defined via matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular, it is completely independent of the construction of the motivic homotopy category. We show that a particular scheme morphism, which exhibits the Jouanolou device as an affine torsor bundle over the projective line, induces a monoid morphism from Cazanave's monoid to this group. Moreover, we show that this monoid morphism is a group completion to a subgroup of the group of scheme morphisms from the Jouanolou device to the projective line. This subgroup is generated by a set of morphisms that are very simple to describe.

引用次数: 0

Multiplication operators on the Bergman space of bounded domains

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110045

Hansong Huang , Dechao Zheng

In this paper we study multiplication operators on Bergman spaces of high dimensional bounded domains and those von Neumann algebras induced by them via the geometry of domains and function theory of their symbols. In particular, using local inverses and

L_{a}^{2}

-removability, we show that for a holomorphic proper map

Φ = (ϕ_{1}, ϕ_{2}, \dots, ϕ_{d})

on a bounded domain Ω in

C^{d}

, the dimension of the von Neumann algebra

V^{⁎} (Φ, Ω)

consisting of bounded operators on the Bergman space

L_{a}^{2} (Ω)

, which commute with both

M_{ϕ_{j}}

and its adjoint

M_{ϕ_{j}}^{⁎}

for each j, equals the number of components of the complex manifold

S_{Φ} = {(z, w) \in Ω^{2} : Φ (z) = Φ (w), z \notin Φ^{- 1} (Φ (Z))}

, where Z is the zero variety of the Jacobian JΦ of Φ. This extends the main result in [14] in high dimensional complex domains. Moreover we show that the von Neumann algebra

V^{⁎} (Φ, Ω)

may not be abelian in general although Douglas, Putinar and Wang [15] showed that

V^{⁎} (Φ, D)

for the unit disk

D

is abelian.

引用次数: 0

Orbifolds and minimal modular extensions

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2025.110103

Chongying Dong , Siu-Hung Ng , Li Ren

<div><div>Let V be a simple vertex operator algebra and G a finite automorphism group of V such that <math><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></math> is regular, and the conformal weight of any irreducible g-twisted V-module N for <math><mi>g</mi><mo>∈</mo><mi>G</mi></math> is nonnegative and is zero if and only if <math><mi>N</mi><mo>=</mo><mi>V</mi></math>. It is established that if V is holomorphic, then the <math><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></math>-module category <math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></mrow></msub></math> is a minimal modular extension of <math><mi>E</mi><mo>=</mo><mrow><mi>Rep</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math>, and is equivalent to the Drinfeld center <math><mi>Z</mi><mo>(</mo><msubsup><mrow><mi>Vec</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>)</mo></math> as modular tensor categories for some <math><mi>α</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math> with a canonical embedding of <math><mi>E</mi></math>. Moreover, the collection <math><msub><mrow><mi>M</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>)</mo></math> of equivalence classes of the minimal modular extensions <math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></mrow></msub></math> of <math><mi>E</mi></math> for holomorphic vertex operator algebras V with a G-action forms a group, which is isomorphic to a subgroup of <math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math>. Furthermore, any pointed modular category <math><mi>Z</mi><mo>(</mo><msubsup><mrow><mi>Vec</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>)</mo></math> is equivalent to <math><msub><mrow><mi>C</mi></mrow><mrow><msubsup><mrow><mi>V</mi></mrow><mrow><mi>L</mi></mrow><mrow><mi>G</mi></mrow></msubsup></mrow></msub></math> for some positive definite even unimodular lattice L. In general, for any rational vertex operator algebra U with a G-action, <math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>U</mi></mrow><mrow><mi>G</mi></mrow></msup></mrow></msub></math> is a minimal modular extension of the braided fusion subcategory <math><mi>F</mi></math> generated by the <math><msup><mrow><mi>U</mi></mrow><mrow><mi>G</mi></mrow></msup></math>-submodules of U-modules. F

引用次数: 0

Maximality of correspondence representations

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110097

Boris Bilich

In this paper, we fully characterize maximal representations of a C*-correspondence, thereby strengthening several earlier results. We demonstrate the maximality criteria through diverse examples. We also describe the noncommutative Choquet boundary and provide additional counterexamples to Arveson's hyperrigidity conjecture following the counterexample recently found by the author and Dor-On. Furthermore, we identify several classes of correspondences for which the hyperrigidity conjecture holds.

引用次数: 0

Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110078

Ruslan Maksimau

We prove the conjecture that flag versions of quiver Grassmannians (also known as Lusztig's fibers) for Dynkin quivers (types A, D, E) have no odd cohomology groups over an arbitrary ring. Moreover, for types A and D we prove that these varieties have affine pavings. We also show that to prove the same statement for type E, it is enough to check this for indecomposable representations.

We also give a flag version of the result of Cerulli Irelli-Esposito-Franzen-Reineke on rigid representations: we prove that flag versions of quiver Grassmannians for rigid representations have a diagonal decomposition. In particular, they have no odd cohomology groups.

引用次数: 0

Bosonization of Feigin-Odesskii Poisson varieties

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-01 DOI: 10.1016/j.aim.2024.110096

Zheng Hua , Alexander Polishchuk

The derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Projective spaces with Feigin-Odesskii Poisson brackets are examples of such moduli spaces over complex elliptic curves [6], [7]. By generalizing several results in our previous work [10], [11], [12] we construct a collection of auxiliary Poisson varieties equipped with Poisson morphisms to Feigin-Odesskii varieties. We call them bosonizations of Feigin-Odesskii varieties. These spaces appear as special cases of the moduli spaces of chains, which we introduce. We show that the moduli space of chains admits a shifted Poisson structure when the base is a Calabi-Yau variety of an arbitrary dimension. Using bosonization spaces mapping to the zero loci of the Feigin-Odesskii varieties, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries. We also derive explicit formulas for the Poisson brackets on the bosonizations of the Feigin-Odesskii varieties associated with line bundles in a simplest nontrivial case.

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引用次数: 0

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