Mathematika最新文献

In this article, we present a new linear independence criterion for values of the $$-adic polygamma functions defined by Diamond. As an application, we obtain the linear independence of some families of values of the $$-adic Hurwitz zeta function $$ at distinct shifts $$. This improves and extends a previous result due to Bel (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) IX (2010), 189–227), as well as irrationality results established by Beukers (Acta Math. Sin. 24 (2008), 663–686). Our proof is based on a novel and explicit construction of Padé-type approximants of the second kind of Diamond's $$-adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.

Let $$ if $$ is the sum of two perfect squares, and $$ otherwise. We study the variance of $$ in short intervals by relating the variance with the second moment of the generating function $$ along $$. We develop a new method for estimating fractional moments of $$-functions and apply it to the second moment of $$ to bound the variance of $$. Our results are conditional on the Riemann hypothesis for the zeta-function and the Dirichlet $$-function associated with the non-principal character modulo 4.

Given two subsets $$ and a binary relation $$, the restricted sumset of $$ with respect to $$ is defined as $$. When $$ is taken as the equality relation, determining the minimum value of $$ is the famous Erdős–Heilbronn problem, which was solved separately by Dias da Silva, Hamidoune and Alon, Nathanson and Ruzsa. Lev later conjectured that if $$ with $$ and $$ is a matching between subsets of $$ and $$, then $$. We confirm this conjecture in the case where $$ for any $$, provided that $$ for some sufficiently large $$ depending only on $$. Our proof builds on a recent work by Bollobás, Leader, and Tiba, and a rectifiability argument developed by Green and Ruzsa. Furthermore, our method extends to cases when $$ is a degree-bounded relation, either on both sides $$ and $$ or solely on the smaller set. In addition, we construct subsets $$ with $$ such that $$ for any prime number $$, where $$ is a matching on $$. This extends an earlier construction by Lev and highlights a distinction between the combinatorial notion of the restricted sumset and the classcial Erdős–Heilbronn problem, where $$ holds given $$ is the equality relation on $$ and $$.

In this work, we study in greater detail than before, J.H. Conway's topographs for integral binary quadratic forms. These are trees in the plane with regions labeled by integers following a simple pattern. Each topograph can display the values of a single form, or represent an equivalence class of forms. We give a new treatment of reduction of forms to canonical equivalence class representatives by employing topographs and a novel continued fraction for complex numbers. This allows uniform reduction for any positive, negative, square, or nonsquare discriminant. Topograph geometry also provides new class number formulas, and short proofs of results of Gauss relating to sums of three squares. Generalizations of the series of Hurwitz for class numbers give evaluations of certain infinite series, summed over the regions or edges of a topograph.

Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius $$, where the endpoints of the chords are drawn according to a given probability distribution on $$. We show that, for $$, the problem is degenerated in the sense that any continuous measure is an extremizer, and that, for $$ sufficiently close to 1, the desired maximal value is strictly below the one for $$ by a polynomial factor in $$. Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is $$ for $$. Connections with other variational problems and energy minimization problems are also presented.

Extending results of Szalay, Bennett, Bugeaud and Mignotte in this paper, we prove finiteness results concerning perfect powers having two or three digits in their representation in a canonical number system of the equation order of an algebraic number field.

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger–Jammes-type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias, and Savo. In addition, we compute the full spectrum in the case of the Euclidean 2-ball and 4-ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the nonmagnetic case due to Colbois, Girouard, and Hassannezhad.

Suppose $$ is a finite abelian group, $$ is not contained in any strict coset in $$, and $$ are dense subsets of $$ such that the sumset $$ avoids $$. We show that $$ and $$ are almost entirely contained in sets defined by a bounded number of coordinates, that is, sets $$ and $$, where the size of $$ is non-zero and independent of $$, and $$ are subsets of $$ such that $$ avoids $$. Furthermore, we show that this result extends to any finite group and $$ summands for any $$.

Let $$, and let $$ be an expanding piecewise linear map sending each interval of linearity to [0,1]. For $$, $$, and $$, we consider the recurrence counting function

We investigate when the local Lipschitz property of the real-valued function $$ implies the global Lipschitz property of the mapping $$ between the metric spaces $$ and $$. Here, $$ denotes the distance of $$ from the non-empty set $$. As a consequence, we find that an analytic function on a uniform domain of a normed space belongs to the Lipschitz class if and only if its modulus satisfies the same condition; in the case of the unit disk this result is proved by Dyakonov. We use the recently established version of a classical theorem by Hardy and Littlewood for mappings between metric spaces. This paper is a continuation of the recent article by the author [Marković, J. Geom. Anal. 34 (2024), https://doi.org/10.48550/arXiv.2405.11509].