Mathematika最新文献

Let $$ be an elliptic curve defined over $$, and let $$ be an imaginary quadratic field. Consider an odd prime $$ at which $$ has good supersingular reduction with $$ and which is inert in $$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell–Weil ranks of $$ are bounded over any subextensions of the anticyclotomic $$-extension of $$. Additionally, we provide an asymptotic formula for the growth of the $$-parts of the Tate–Shafarevich groups of $$ over these extensions.

We prove the direct and the converse inequalities for type IV superorthogonality in the vector-valued setting. The converse one is also new in the scalar setting.

We study the $$ moment of central values of the family of Dirichlet $$-functions to a fixed prime modulus and establish sharp upper bounds for all real $$.

In Euclidean space, the asymptotic shape of large cells in various types of Poisson-driven random tessellations has been the subject of a famous conjecture due to David Kendall. Since shape is a geometric concept and large cells are identified by means of geometric size functionals, the resolution of the conjecture is inevitably connected with geometric inequalities of isoperimetric type and their improvements in the form of geometric stability results, relating geometric size functionals and hitting functionals. The latter are deterministic characteristics of the underlying random tessellation. The current work explores specific and typical cells of random tessellations in spherical space. A key ingredient of our approach is new geometric inequalities and quantitative strengthenings in terms of stability results for general and also for some specific size and hitting functionals of spherically convex bodies. As a consequence, we obtain probabilistic deviation inequalities and asymptotic distributions of quite general size functionals. In contrast to the Euclidean setting, where naturally the asymptotic regime concerns large size, in the spherical framework, the asymptotic analysis is primarily concerned with high intensities.

In this article, we study discrete maximal function associated with the Birch–Magyar averages over sparse sequences. We establish sparse domination principle for such operators. As a consequence, we obtain $$-estimates for such discrete maximal functions over sparse sequences for all $$. The proof of sparse bounds is based on scale-free $$-improving estimates for the single scale Birch–Magyar averages.

Three new combinations of convex bodies are introduced and studied: the $$ fiber, $$ chord, and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation generalizes the classical Steiner symmetral, albeit in different ways. For the $$ fiber and $$ chord combinations, we derive Brunn–Minkowski-type inequalities and the corresponding Minkowski's first inequalities. We also prove that the general affine surface areas are concave (respectively, convex) with respect to the graph sum, thereby generalizing fundamental results of Ye (Indiana Univ. Math. J. 14 (2014), 1–19) on the monotonicity of the general affine surface areas under Steiner symmetrization. As an application, we deduce a corresponding Minkowski's first inequality for the $$ affine surface area of a graph combination of convex bodies.

Motivated by general probability theory, we say that the set $$ in $$ is antipodal of rank $$, if for any $$ elements $$, there is an affine map from $$ to the $$-dimensional simplex $$ that maps $$ bijectively onto the $$ vertices of $$. For $$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank $$ in $$? We present a geometric characterization of antipodal sets of rank $$ and adapting the argument of Danzer and Grünbaum originally developed for the $$ case, we prove an upper bound which is exponential in the dimension. We show that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension. By connecting rank-$$ antipodality to $$-neighborly polytopes, we obtain another upper bound when $$.

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 $$.