# Laboratoire IMATH

Institut de Mathématiques de Toulon (EA 2134)

## Séminaire de Alessandro DELLA CORTE (Memocs)

Séminaire MN-AA
jeudi 19/09/19, 14h, salle M005

Kolakoski sequence : open problems and new approaches

Résumé : The Kolakoski sequence S is the unique element of $1,2^\mathbbN$ starting with $1$ and coinciding with its own run length encoding :

S = 122112122122112…

A famous conjecture by Keane asks whether the density of 1s exists and equals 1/2, while questions concerning the recurrence, the mirror invariance and the transitivity of the sequence are also open since the ‘90s.

In the talk some new approaches will be proposed, and in particular some logical connections between reversal invariance, mirror invariance and recurrence will be shown, as well as a uniformity result for the convergence of the frequency to the asymptotic density for words of arbitrary length.