We study the irreducible constituents of the reduction modulo of irreducible algebraic representations of the group for a finite extension of . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of and the central character of its reduction modulo . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.
We study the asymptotic behaviour of the summatory function of a class of arithmetic functions. These functions are generalizations of the well-known general 4-dimensional divisor function d₄(n). We show that the corresponding error estimate is the best one can obtain by the present methods of analytic number theory.
We establish necessary and sufficient conditions for the convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. The multiplicative measures depict distributions of component spectra of random structures and also the equilibria of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We show that the convergence of multiplicative measures is equivalent to the asymptotic independence of counts of...
We study the asymptotics conjecture of Malle for dihedral groups of order , where is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen–Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.
The variance of the number of lattice points inside the dilated bounded set with random position in has asymptotics if the rotational average of the squared modulus of the Fourier transform of the set is . The asymptotics follow from Wiener’s Tauberian theorem.
Nous étudions le comportement asymptotique d’une classe de suites mahlériennes dont les séries génératrices sont des produits infinis. Un exemple caractéristique est celui de l’estimation des coefficients de Taylor de , voisin des partitions binaires étudiées par De Bruijn. Le résultat obtenu illustre un cas typique d’une classification naturelle des suites mahlériennes. Les techniques utilisées, transformation de Mellin ou méthode du col, ressortissent à la théorie analytique des nombres et à...
The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that...