In order to understand problems in dynamics which are sensitive to
arithmetic properties of return times to regions, it is desirable to
generalize classical results about rotations on the circle to the
setting of rotations on adelic tori. One such result is the classical
three gap theorem, which is also referred to as the three distance
theorem and as the Steinhaus problem. It states that, for any real
number, a, and positive integer, N, the collection of points na mod 1,
where n runs from 1 to N, partitions the circle into component arcs
having one of at most three distinct lengths. Since the 1950s, when
this theorem was first proved independently by multiple authors, it
has been reproved numerous times and generalized in many ways. One of
the more recent proofs has been given by Marklof and Strömbergsson
using a lattice based approach to gaps problems in Diophantine
approximation. In this talk, we use an adaptation of this approach to
the adeles to prove a natural generalization of the classical three
gap theorem for rotations on adelic tori. This is joint work with Alan
Haynes.
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