  The three gap theorem, also known as the 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 Erd\H{o}s, 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. Our results in
particular improve earlier work by Boshernitzan, Dyson and Bleher et al. We
furthermore discuss a close link with the Littlewood conjecture in
multiplicative Diophantine approximation. Finally, we also demonstrate how our
methods can be adapted to obtain similar results for gaps between return times
of translations to shrinking regions on higher dimensional tori.
