Volume 37, Issue 2

The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in 𝔽 q 2 {\mathbb{F}_{q}^{2}} . More precisely, for A , B , C ⊂ 𝔽 q 2 {A,B,C\subset\mathbb{F}_{q}^{2}} , if | A | ⁢ | B | ⁢ | C | 1 2 ≫ q 4 {|A||B||C|^{\frac{1}{2}}\gg q^{4}} , then for any λ ∈ 𝔽 q ∖ { 0 } {\lambda\in\mathbb{F}_{q}\setminus\{0\}} , the number of congruence classes of triangles with vertices in A × B × C {A\times B\times C} and one side-length λ is at least ≫ q 2 {\gg q^{2}} . In higher dimensions, we obtain similar results for k -simplex but under a slightly stronger condition. Compared to the well-known L 2 {L^{2}} method in the literature, our approach offers better results in both conditions and conclusions. When A = B = C {A=B=C} , the second goal of this paper is to give a new and unified proof of the best current results on the distribution of simplex due to Bennett, Hart, Iosevich, Pakianathan and Rudnev (2017) and McDonald (2020). The third goal of this paper is to study a Furstenberg-type problem associated to a set of rigid motions. The main ingredients in our proofs are incidence bounds between points and rigid motions. While the incidence bounds for large sets are due to the author and Semin Yoo (2023), the bound for small sets will be proved by using a point–line incidence bound in 𝔽 q 3 {\mathbb{F}_{q}^{3}} due to Kollár (2015).

This paper concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional p -Laplace operator in ℝ N {\mathbb{R}^{N}} . Precisely, we study the Kirchhoff-type problem ( a + b ⁢ ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ d x ⁢ d y ) ⁢ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = f ⁢ ( x , u )   in  ⁢ ℝ N , \Biggl{(}a+b\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,% \mathrm{d}x\,\mathrm{d}y\Biggr{)}(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u)\quad% \text{in }\mathbb{R}^{N}, where a , b > 0 {a,b>0} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p -Laplacian with 0 < s < 1 < p < N s {0<s<1<p<\frac{N}{s}} , V : ℝ N → ℝ {V\colon\mathbb{R}^{N}\to\mathbb{R}} and f : ℝ N × ℝ → ℝ {f\colon\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}} are continuous functions while V can have negative values and f fulfills suitable growth assumptions. According to the interaction between the attenuation of the potential at infinity and the behavior of the nonlinear term at the origin, using a penalization argument along with L ∞ {L^{\infty}} -estimates and variational methods, we prove the existence of a positive solution. In addition, we also establish the existence of infinitely many solutions provided the nonlinear term is odd.

In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight- k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

We give some q -supercongruences from a q -analogue of Watson’s F 2 3 {{}_{3}F_{2}} summation and the method of “creative microscoping”, introduced by the author and Zudilin. These q -supercongruences may be considered as further generalizations of the (A.2) supercongruence of Van Hamme modulo p 3 {p^{3}} or p 2 {p^{2}} for any odd prime p . Meanwhile, we confirm a supercongruence conjecture of Wang and Yue through establishing its q -analogue.

Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.

Let A and B be two unipotent elements of SU ⁢ ( 2 , 1 ) {\mathrm{SU}(2,1)} with distinct fixed points. In [S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1, Paper No. 9], the authors gave several conditions that guarantee the subgroup 〈 A , B 〉 {\langle A,B\rangle} is discrete and free by using Klein’s combination theorem. We will improve their conditions by using a variant of Klein’s combination theorem. With the same arguments and the additional assumption that AB is unipotent, we also extend Parker and Will’s condition that guarantees the subgroup 〈 A , B 〉 {\langle A,B\rangle} is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].

In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

In 2004, Thakur introduced a positive characteristic analogue of multizeta values. Later, in 2017, he mentioned the two colored variants which are positive characteristic analogues of colored multizeta values in his survey of multizeta values in positive characteristic. In this paper, we study one of those two variants. We establish their fundamental properties, that include their non-vanishing, sum-shuffle relations, 𝑡-motivic interpretation and linear independence. For the linear independence results, we prove that there are no nontrivial k ̄ \overline{k} -linear relations among the colored multizeta values with different weights.

For a given normalized newform f of large prime level, we establish a lower bound with respect to the level for the number of normalized newforms g of the same weight and level as of f such that the central L -values of f and g both twisted by a quadratic character do not vanish.

Let G ⁢ ( g ; x ) := ∑ n ≤ x g ⁢ ( n ) {G(g;x):=\sum_{n\leq x}g(n)} be the summatory function of an arithmetical function g ⁢ ( n ) {g(n)} . In this paper, we prove that we can write weighted averages of an arbitrary fixed number N of arithmetical functions g j ⁢ ( n ) , j ∈ { 1 , … , N } {g_{j}(n),\,j\in\{1,\dots,N\}} as an integral involving the convolution (in the sense of Laplace) of G j ⁢ ( x ) {G_{j}(x)} , j ∈ { 1 , … , N } {j\in\{1,\dots,N\}} . Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.

Let p be a prime number. One of the most difficult and important questions in Galois theory these days is determining which pro- p groups can occur as maximal pro- p Galois groups. Some restrictions of the structure of maximal pro- p Galois groups are already known, and therefore, the families of all pro- p groups satisfying these restrictions are in the center of the research. In the current paper we deal with three of these families: The family of all pro- p groups which satisfy hereditarily the n -vanishing Massey product property, the family of 1-cyclotomic oriented pro- p groups, and the family of pro- p groups that are hereditarily of p -absolute Galois type. We show that all these families are closed under taking profinite completion of each order.

A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of K ⁢ ( z , z ) {K(z,z)} (where K ⁢ ( z , w ) {K(z,w)} is an automorphic kernel function) over the fundamental domain, one considers the integral of K ⁢ ( z , z ) ⁢ u ⁢ ( z ) {K(z,z)u(z)} , where u ⁢ ( z ) {u(z)} is a fixed automorphic eigenfunction of the Laplace operator. This formula was proved for discrete subgroups of PSL ⁢ ( 2 , ℝ ) {\mathrm{PSL}(2,\mathbb{R})} , and just as in the case of the classical Selberg trace formula it was obtained by evaluating in two different ways (“geometrically” and “spectrally”) the integral of K ⁢ ( z , z ) ⁢ u ⁢ ( z ) {K(z,z)u(z)} . In the present paper we work out the geometric side of a further generalization of this generalized trace formula: we consider the case of discrete subgroups of PSL ⁢ ( 2 , ℝ ) n {\mathrm{PSL}(2,\mathbb{R})^{n}} where n > 1 {n>1} . Many new difficulties arise in the case of these groups due to the fact that the classification of conjugacy classes is much more complicated for n > 1 {n>1} than in the case n = 1 {n=1} .

We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive ∃ ∀ {\exists\forall} -sentence true in 𝐅 3 {\mathbf{F}_{3}} and false in 𝐅 4 {\mathbf{F}_{4}} . Secondly, we show that every model of Th ⁢ ( 𝐅 n ) {\mathrm{Th}(\mathbf{F}_{n})} admits a canonical homomorphism into the profinite-bounded completion 𝐇 n {\mathbf{H}_{n}} of 𝐅 n {\mathbf{F}_{n}} . Thirdly, we show that 𝐇 n {\mathbf{H}_{n}} is isomorphic to the Dedekind–MacNeille completion of 𝐅 n {\mathbf{F}_{n}} , and that 𝐇 n {\mathbf{H}_{n}} is not positively elementarily equivalent to 𝐅 n {\mathbf{F}_{n}} , as there is a positive ∀ ∃ {\forall\exists} -sentence true in 𝐇 n {\mathbf{H}_{n}} and false in 𝐅 n {\mathbf{F}_{n}} . Finally, we show that DM ⁢ ( 𝐅 n ) {\mathrm{DM}(\mathbf{F}_{n})} is a retract of Id ⁢ ( 𝐅 n ) {\mathrm{Id}(\mathbf{F}_{n})} and that for any lattice 𝐊 {\mathbf{K}} which satisfies Whitman’s condition ( W ) {\mathrm{(W)}} and which is generated by join prime elements, the three lattices 𝐊 {\mathbf{K}} , DM ⁢ ( 𝐊 ) {\mathrm{DM}(\mathbf{K})} , and Id ⁢ ( 𝐊 ) {\mathrm{Id}(\mathbf{K})} all share the same positive universal first-order theory.

An RD-space 𝑀 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝑀. In this paper, firstly, the authors give the Plancherel–Pôlya characterization of product weighted Triebel–Lizorkin spaces and product weighted Besov spaces on RD-spaces and make some estimates for the product singular integral operators in Journé’s class on these function spaces. As a result of these conclusions, they present some sufficient conditions for the boundedness of product singular integral operators on the product Lipschitz spaces and product weighted Hardy spaces. Secondly, by the boundedness of lifting and projection operators, they also obtain that the dual spaces of the product weighted Hardy spaces are product weighted Carleson measure spaces. Using the idea of dual, the authors obtain the weighted boundedness of singular integral operators on the product weighted Carleson measure spaces.

We show that for each abelian number field K of sufficiently large degree d there exists an element α ∈ K {\alpha\in K} with K = ℚ ⁢ ( α ) {K=\mathbb{Q}(\alpha)} and absolute Weil height H ⁢ ( α ) ≪ d | Δ K | 1 2 ⁢ d {H(\alpha)\ll_{d}|\Delta_{K}|^{\frac{1}{2d}}} , where Δ K {\Delta_{K}} denotes the discriminant of K . This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent 1 2 ⁢ d {\frac{1}{2d}} is best-possible when d is even.

This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 L^{2} -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 n\geq 3 .

The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved.