Volume 35, Issue 2

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for X 2 3 + ϵ < H < X 1 - ϵ {X^{\frac{2}{3}+\epsilon}<H<X^{1-\epsilon}} there are constants B h {B_{h}} such that ∑ X ≤ n ≤ 2 ⁢ X λ f ⁢ ( n ) 2 ⁢ λ f ⁢ ( n + h ) 2 - B h ⁢ X = O f , A , ϵ ⁢ ( X ⁢ ( log ⁡ X ) - A ) \sum_{X\leq n\leq 2X}\lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,% \epsilon}(X(\log X)^{-A}) for all but O f , A , ϵ ⁢ ( H ⁢ ( log ⁡ X ) - 3 ⁢ A ) {O_{f,A,\epsilon}(H(\log X)^{-3A})} integers h ∈ [ 1 , H ] {h\in[1,H]} where { λ f ⁢ ( n ) } n ≥ 1 {\{\lambda_{f}(n)\}_{n\geq 1}} are normalized Hecke eigenvalues of a fixed holomorphic cusp form f . Our method is based on the Hardy–Littlewood circle method. We divide the minor arcs into two parts m 1 {m_{1}} and m 2 {m_{2}} . In order to treat m 2 {m_{2}} , we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matomäki, Radziwiłł and Tao. In order to treat m 1 {m_{1}} , we apply Parseval’s identity and Gallagher’s lemma.

We show that the splitting conjecture in the hybrid model of Gonek, Hughes and Keating holds to order on the Riemann hypothesis. Our results are valid in a larger range of the parameter X which mediates between the partial Euler and Hadamard products. We also show that the asymptotic splitting conjecture holds for this larger range of X in the cases of the second and fourth moments.

We study almost Dedekind domains with respect to the failure of ideals to have radical factorization, that is, we study how to measure how far an almost Dedekind domain is from being an SP-domain. To do so, we consider the maximal space ℳ = Max ⁡ ( R ) {\mathcal{M}=\operatorname{Max}(R)} of an almost Dedekind domain R , interpreting its (fractional) ideals as maps from ℳ {\mathcal{M}} to ℤ {\mathbb{Z}} , and looking at the continuity of these maps when ℳ {\mathcal{M}} is endowed with the inverse topology and ℤ {\mathbb{Z}} with the discrete topology. We generalize the concept of critical ideals by introducing a well-ordered chain of closed subsets of ℳ {\mathcal{M}} (of which the set of critical ideals is the first step) and use it to define the class of SP-scattered domains , which includes the almost Dedekind domains such that ℳ {\mathcal{M}} is scattered and, in particular, the almost Dedekind domains such that ℳ {\mathcal{M}} is countable. We show that for this class of rings the group Inv ⁢ ( R ) {\mathrm{Inv}(R)} is free by expressing it as a direct sum of groups of continuous maps, and that, for every length function ℓ {\ell} on R and every ideal I of R , the length of R / I {R/I} is equal to the length of R / rad ⁡ ( I ) {R/{\operatorname{rad}(I)}} .

We investigate the structure of finite group G in which the centralizer of each non-central primary element of G is maximal in G . This provides an answer to the question raised in [X. Zhao, R. Chen and X. Guo, Groups in which the centralizer of any non-central element is maximal, J. Group Theory 23 2020, 5, 871–878], and also is a generalization of the result in [A. Mann, Finite groups with maximal normalizers, Illinois J. Math. 12 1968, 67–75]. In particular, we give an independent criterion of simplicity of a group, asserting that a group G is simple if the centralizer of each non-central primary element is maximal in G .

We consider compact manifolds M with a cohomogeneity one action of a compact Lie group G such that the orbit space M / G {M/G} is a closed interval. For T a maximal torus of G , we find necessary and sufficient conditions on the group diagram of M such that the T -action on M is of GKM type, and describe its GKM graph. The general results are illustrated on explicit examples.

The Gruenberg–Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices p , q are joined by an edge whenever the group has an element of order pq . It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well-studied class of rational groups, and has received attention recently. In the first part of this paper, we give a complete classification of the Gruenberg–Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg–Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg–Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg–Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups for the classes of cut groups and rational groups, respectively. The prime graph question asks whether the Gruenberg–Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the first Zassenhaus conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the prime graph question for integral group rings for finite rational groups and most finite cut groups

In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner identity, and leads to “semilinear Calderón–Zygmund” type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclidean space will also be discussed.

We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form min ⁡ { ∫ Ω F ⁢ ( x , D ⁢ w ) ⁢ 𝑑 x : w ∈ 𝒦 ψ ⁢ ( Ω ) } , \min\bigg{\{}\int_{\Omega}F(x,Dw)dx:w\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where Ω is a bounded open set of ℝ n {\mathbb{R}^{n}} , n ≥ 2 {n\geq 2} , the function ψ ∈ W 1 , p ⁢ ( Ω ) {\psi\in W^{1,p}(\Omega)} is a fixed function called obstacle, and 𝒦 ψ ⁢ ( Ω ) := { w ∈ W 1 , p ⁢ ( Ω ) : w ≥ ψ ⁢  a.e. in  ⁢ Ω } \mathcal{K}_{\psi}(\Omega):=\bigl{\{}w\in W^{1,p}(\Omega):w\geq\psi\text{ a.e.% in }\Omega\bigr{\}} is the class of admissible functions. If the obstacle ψ is locally bounded, we prove that the gradient of solution inherits some fractional differentiability property, assuming that both the gradient of the obstacle and the mapping x ↦ D ξ ⁢ F ⁢ ( x , ξ ) {x\mapsto D_{\xi}F(x,\xi)} belong to some suitable Besov space. The main novelty is that the Besov type regularity on the partial map x ↦ D ξ ⁢ F ⁢ ( x , ξ ) {x\mapsto D_{\xi}F(x,\xi)} is not related to the dimension n .

We study interpolation of the measure of noncompactness of homogeneous polynomials on Banach spaces. We prove that, for a large class of interpolation functors, preserving interpolation of measures of noncompactness of interpolated linear operators between Banach couples can be lifted to polynomials. As an application, we show that the measure of noncompactness of polynomials behaves well under the real method of interpolation.

Let R be any ring with identity. We show that the homotopy category of all acyclic chain complexes of pure-projective R -modules is a compactly generated triangulated category. We do this by constructing abelian model structures that put this homotopy category into a recollement with two other compactly generated triangulated categories: The usual derived category of R and the pure derived category of R .

Hecke–Kiselman algebras A Θ {A_{\Theta}} , over a field 𝕂 {\mathbb{K}} , associated to finite oriented graphs Θ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand–Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph Θ that determines the dimension of A Θ {A_{\Theta}} is found. Namely, we prove that the Gelfand–Kirillov dimension of A Θ {A_{\Theta}} is the sum of the number of cyclic subgraphs of Θ and the number of oriented paths of a special type in the graph, each counted certain specific number of times.

We study the Li–Yau inequality for the heat equation corresponding to the Dunkl harmonic oscillator, which is a nonlocal Schrödinger operator parameterized by reflections and multiplicity functions. In the particular case when the reflection group is isomorphic to ℤ 2 d {\mathbb{Z}_{2}^{d}} , the result is sharp in the sense that equality is achieved by the heat kernel of the classic harmonic oscillator. We also provide the application on parabolic Harnack inequalities.

This paper takes the first steps towards a systematic study of additive rigid meromorphic cocycles of higher weight. These were introduced by Darmon and Vonk, who focused on additive cocycles of weight two and their multiplicative lifts. After classifying certain rigid meromorphic cocycles of weight 2 ⁢ k {2k} , we construct an explicit holomorphic kernel function realizing a Shimura–Shintani style correspondence from modular forms of weight k + 1 / 2 {k+1/2} and level 4 ⁢ p 2 {4p^{2}} to rigid analytic cocycles of weight 2 ⁢ k {2k} on SL 2 ⁡ ( ℤ ⁢ [ 1 / p ] ) {\operatorname{SL}_{2}(\mathbb{Z}[1/p])} .

Motivated by a classic result for free groups, one says that a group G has the Magnus property if the following holds: Whenever two elements generate the same normal subgroup of G , they are conjugate or inverse-conjugate in G . It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most 2. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishing and for disproving the property. We also prove that a free center-by-(polynilpotent of given class row) group has the Magnus property if and only if it is nilpotent of class at most 2. On the way, we display 2-generated nilpotent groups (with non-trivial torsion) of any prescribed nilpotency class with the Magnus property. Similar examples of finitely generated, torsion-free nilpotent groups are hard to come by, but we construct a 4-generated, torsion-free, class-3 nilpotent group of Hirsch length 9 with the Magnus property. Furthermore, using a weak variant of the Magnus property and an ultraproduct construction, we establish the existence of metabelian, torsion-free, nilpotent groups of any prescribed nilpotency class with the Magnus property.