|Dept.||University of Houston|
The Steinhaus theorem, known colloquially as the 3-distance theorem, states that for any positive integer N and for any real number x, the collection of points nx modulo 1, with 0<n<N, partitions R/Z into component intervals which each have one of at most 3 possible distinct lengths. Many authors have explored higher dimensional generalizations of this theorem. In this talk we will survey some of their results, and we will explore a two-dimensional version of the problem, which turns out to be closely related to the Littlewood conjecture. We will explain how tools from homogeneous dynamics can be employed to obtain new results about a problem of Erdos and Geelen and Simpson, proving the existence of parameters for which the number of distinct gaps in a higher dimensional version of the Steinhaus problem is unbounded. This is joint work with Jens Marklof.