Speaker: Alan Haynes
11 November, 2020 13:30 UTC
Abstract: The three distance theorem states that, if $x$ is any real number and $N$ is any positive integer, the points $x, 2x, … , Nx \mod 1$ partition the unit interval into component intervals having at most $3$ distinct lengths. There are many higher dimensional analogues of this theorem, and in this talk we will discuss two of them. In the first we consider points of the form $mx+ny \mod 1$, where $x$ and $y$ are real numbers and $m$ and $n$ are integers taken from an expanding set in the plane. This version of the problem was previously studied by Geelen and Simpson, Chevallier, Erdős, and many other people, and it is closely related to the Littlewood conjecture in Diophantine approximation. The second version of the problem is a straightforward generalization to rotations on higher dimensional tori which, surprisingly, has been largely overlooked in the literature. For the two dimensional torus, we are able to prove a five distance theorem, which is best possible. In higher dimensions we also have bounds, but establishing optimal bounds is an open problem. The first hour of this talk will be expository, and the second half will focus on proofs. The new results presented in this talk are joint work with Jens Marklof and with Roland Roeder.