Diophantine Approximation in p-adic Solenoids

We discuss problems related to Diophantine approximation in p-adic solenoids and their interactions with dynamical systems.

Bounded remainder sets for rotations on the real torus, in all dimesnions, have been widely studied since the 1920's and there now exists a complete classification of volumes of bounded remainder sets for toral rotations in any dimension. Subsequently, a classification of bounded remainder sets for rotations on an uncountable collection of connected, compact subgroups of the adelic torus has been given.

We complete this direction of inquiry and give an explicit construction of polytopal bounded remainder sets of all possible volumes, for any ergodic rotation on the adelic torus in any dimesnion. We prove a necessary and sufficient condition for ergodicity of the rotation and show that ergodicity in this setting is equivalent to unique ergodicity. We also relate the existence of bounded remainder sets to the existence of dynamical coboundaries for the rotation map. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument that reduces the existence argument to the case of an irrational rotation on the d-dimensional real torus. We also verify that all allowable volumes are obtained by this construction.

The classical three gap theorem for rotations on the unit circle was first proved in the late 1950's and since then it has been reproved numerous times and generalized in many ways. In order to understand problems in dynamics that are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on the real torus to the setting of rotations on adelic tori.

We state and prove a natural generalization of the Three Gap Theorem for rotations on adelic tori. Our proof uses an adaptation of a lattice based approach to gaps problems in Diophantine approximation. We reformulate our problem as a problem about bounding a certain function on a space of lattices. We also give an exhaustive list of examples to prove that the bound we get is best possible.