Advances in Mathematics最新文献

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On primes in arithmetic progressions and bounded gaps between many primes

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.aim.2025.110190

Julia Stadlmann

We prove that the primes below x are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to

x^{1 / 2 + 1 / 40 - ϵ}

. The exponent of distribution

\frac{1}{2} + \frac{1}{40}

improves on a result of Polymath [13], who had previously obtained the exponent

\frac{1}{2} + \frac{7}{300}

. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that

\underset{n \to \infty}{\lim \inf} (p_{n + m} - p_{n}) = O (\exp (3.8075 m)) .

The main new ingredient of our proof is a modification of the q-van der Corput process. It allows us to exploit additional averaging for the exponential sums which appear in the Type I estimates of [13].

引用次数: 0

Kneser graphs are Hamiltonian

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.aim.2025.110189

Arturo Merino , Torsten Mütze , Namrata

For integers

k \geq 1

and

n \geq 2 k + 1

, the Kneser graph

K (n, k)

has as vertices all k-element subsets of an n-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph

K (5, 2)

. This problem received considerable attention in the literature, including a recent solution for the sparsest case

n = 2 k + 1

. The main contribution of this paper is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph

J (n, k, s)

has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly s. Clearly, we have

K (n, k) = J (n, k, 0)

, i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known natural families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász' conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway's Game of Life, and to analyze this system combinatorially and via linear algebra.

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引用次数: 0

Constructing smoothings of stable maps

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-03-06 DOI: 10.1016/j.aim.2025.110188

Fatemeh Rezaee , Mohan Swaminathan

Let X be a smooth projective variety. Define a stable map

f : C \to X

to be eventually smoothable if there is an embedding

X ↪ P^{N}

such that

(C, f)

occurs as the limit of a 1-parameter family of stable maps to

P^{N}

with smooth domain curves. Via an explicit deformation-theoretic construction, we produce a large class of stable maps (called stable maps with model ghosts), and show that they are eventually smoothable.

设 X 是光滑射影变种。如果存在一个嵌入 XPN，使得（C,f）作为具有光滑域曲线的稳定映射 PN 的 1 参数族的极限出现，则定义稳定映射 f:C→X 为最终可光滑映射。通过明确的变形理论构造，我们产生了一大类稳定映射（称为具有模型幽灵的稳定映射），并证明它们最终是可平滑的。

引用次数: 0

Non-smoothable homeomorphisms of 4-manifolds with boundary

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-03-06 DOI: 10.1016/j.aim.2025.110191

Daniel Galvin , Roberto Ladu

We construct the first examples of non-smoothable self-homeomorphisms of smooth 4-manifolds with boundary that fix the boundary and act trivially on homology. As a corollary, we construct self-diffeomorphisms of 4-manifolds with boundary that fix the boundary and act trivially on homology but cannot be isotoped to any self-diffeomorphism supported in a collar of the boundary and, in particular, are not isotopic to any generalised Dehn twist.

引用次数: 0

Affine dual Minkowski problems

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-03-05 DOI: 10.1016/j.aim.2025.110184

Xiaxing Cai, Gangsong Leng, Yuchi Wu, Dongmeng Xi

While affine functionals of convex bodies and their affine isoperimetric inequalities have been extensively studied, the construction of geometric measures arising from affine geometric invariants (other than volume) has been missing.

In this work, affine ‘‘invariant’’ measures derived from the dual affine quermassintegrals are presented. Minkowski problems for the new affine-invariant measures are proposed and studied. The new variation formula derived here leads to new affine operators that map star bodies to star bodies. An affine isoperimetric inequality is obtained for new bi-dual intersection bodies.

引用次数: 0

Homological n-systole in (n + 1)-manifolds and bi-Ricci curvature

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-03-03 DOI: 10.1016/j.aim.2025.110187

Jianchun Chu , Man-Chun Lee , Jintian Zhu

In this paper, we prove an optimal systolic inequality and the corresponding rigidity in the equality case on closed manifolds with positive bi-Ricci curvature, which generalizes the work of Bray-Brendle-Neves in [3]. The proof is given in all dimensions based on the method of minimal surfaces under the Generic Regularity Hypothesis, which is known to be true up to dimension ten.

引用次数: 0

On exponential frames near the critical density

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-03-03 DOI: 10.1016/j.aim.2025.110180

Marcin Bownik , Jordy Timo van Velthoven

Given a relatively compact set

Ω \subseteq R

of Lebesgue measure

| Ω |

and

ε > 0

, we show the existence of a set

Λ \subseteq R

of uniform density

D (Λ) \leq (1 + ε) | Ω |

such that the exponential system

{\exp (2 π i λ \cdot) 1_{Ω} : λ \in Λ}

is a frame for

L^{2} (Ω)

with frame bounds

A | Ω |, B | Ω |

for constants

A, B

only depending on ε. This solves a problem on the frame bounds of an exponential frame near the critical density posed by Nitzan, Olevskii and Ulanovskii. We also prove an extension to locally compact abelian groups, which improves a result by Agora, Antezana and Cabrelli by providing frame bounds involving the spectrum.

引用次数: 0

Level set estimates for the periodic Schrödinger maximal function on T1

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-28 DOI: 10.1016/j.aim.2025.110186

Ciprian Demeter

We prove (essentially) sharp

L^{4}

level set estimates for the periodic Schrödinger maximal operator in a certain range of the cut-off parameter.

引用次数: 0

Sequence entropy and IT-tuples for minimal group actions

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-28 DOI: 10.1016/j.aim.2025.110183

Chunlin Liu , Xiangtong Wang , Leiye Xu

Let G be an infinite discrete countable group and

(X, G)

a minimal G-system. First, we prove that

h_{t o p}^{⁎} (X, G) \geq \log \sum_{μ \in M^{e} (X, G)} e^{h_{μ}^{⁎} (X, G)},

where

h_{t o p}^{⁎} (X, G)

and

h_{μ}^{⁎} (X, G)

are the supremum of the topological and metric sequence entropy, respectively. Additionally, if G is abelian, there exists

K \in N \cup {\infty}

with

\log K \leq h_{t o p}^{⁎} (X, G)

such that it is a regular K-to-one extension of its maximal equicontinuous factor.

Furthermore, for any infinite countable discrete group G, we show that if the factor map from a minimal G-system to its maximal equicontinuous factor is regular

K_{1}

-to-one and almost

K_{2}

-to-one, then the system admits

⌈ K_{1} / K_{2} ⌉

-IT-tuples, where

K_{1} \in N \cup {\infty}

and

K_{2} \in N

. As a corollary, we refine the upper bound on the number of ergodic measures for systems that are almost N-to-one extensions of their maximal equicontinuous factors and lack K-IT-tuples, thereby improving the result of Huang et al. (2021) [17].

引用次数: 0

Superspace coinvariants and hyperplane arrangements

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-02-28 DOI: 10.1016/j.aim.2025.110185

Robert Angarone , Patricia Commins , Trevor Karn , Satoshi Murai , Brendon Rhoades

Let Ω be the superspace ring of polynomial-valued differential forms on affine n-space. The natural action of the symmetric group

S_{n}

on n-space induces an action of

S_{n}

on Ω. The superspace coinvariant ring is the quotient SR of Ω by the ideal generated by

S_{n}

-invariants with vanishing constant term. We give the first explicit basis of SR, proving a conjecture of Sagan and Swanson. Our techniques use the theory of hyperplane arrangements. We relate SR to instances of the Solomon–Terao algebras of Abe–Maeno–Murai–Numata and use exact sequences relating the derivation modules of certain ‘southwest closed’ arrangements to obtain the desired basis of SR.

引用次数: 0

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