# Monday 30 Apr 2018: Dynamics Seminar: Higher dimensional Steinhaus problems via homogeneous dynamics

### Jens Marklof - Bristol

**H101** 14:30-15:30

The three gap theorem, also known as Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of \alpha,2\alpha,\ldots, N\alpha take at most three distinct values. Motivated by a question of Erdos, Geelen and Simpson, we explore a higher-dimensional variant, which asks for the number of gaps between the fractional parts of a linear form. Using the ergodic properties of the diagonal action on the space of lattices, we prove that for almost all parameter values the number of distinct gaps in the higher dimensional problem is unbounded. This in particular improves earlier work by Boshernitzan, Dyson and Bleher et al. Joint work with Alan Haynes (Houston).