Archives des manifestations scientifiques

Orateur: Simon Kristensen (Aarhus)

Titre: Partitions and Fibonacci numbers.

Résumé: The asymptotic theory of partitions is classical, going back to (at least) Hardy and Ramanujan who found an asymptotic formula for the number of unrestricted partitions of n. If the partitions are restricted, the problem becomes a different one, and as opposed to the unrestricted case, the asymptotic behaviour can be a more difficult oscillatory function. In this talk, we will find the asymptotic behaviour of the function p_F(n), which counts the number of representations of n as the sum of possibly non-distinct Fibonacci numbers. This function has this oscillatory behaviour. We will also show how this generalises to a variety of linear recurrence sequences. The work is joint with Michael Coons and Mathias L. Laursen.

Oratrice: Maiken Gravgaard (Aarhus)

Titre: Twisted inhomogeneous Dirichlet improbability.

Résumé: For a real valued non-increasing function psi a pair (A, b) consisting of a real valued matrix A with m rows and n columns and a real vector b is called psi-Dirichlet improvable if the system ||Aq+b||^m < psi(T) and |q|^n < T has a solution q in Z^m for all sufficiently large T. Kleinbock and Wadleigh proved a zero-full law for the Lebesgue measure of the set of psi-Dirichlet pairs. 

Kim and Kim established results for the Hausdorff measure of the set of psi-Dirichlet non-improvable pairs as well as for the b-fixed singly metric case. 

Previous proofs have used homogeneous dynamics. In this talk, we show full Hausdorff dimension in the A-fixed singly metric case for all A and certain psi using a different approach to the problem. This is joint with Simon Kristensen.