Abstract
The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of α, 2α, . . . , Nα take at most three distinct values. Motivated by a question of Erdős, Geelen and Simpson, we explore a higherdimensional 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.
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.
Original language  English 

Journal  Annales Scientifiques de l'École Normale Supérieure 
Publication status  Accepted/In press  6 Jul 2018 
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Profiles

Professor Jens Marklof
 Science Faculty Office  Dean of the Faculty of Science and Professor of Mathematical Physics
 Probability, Analysis and Dynamics
 Pure Mathematics
 Ergodic theory and dynamical systems
Person: Academic , Member