The Steinhaus theorem, known colloquially as the 3-distance theorem,
states that for any positive integer \(N\) and for any real number
\(x\), the collection of points \(nx\) modulo 1, with \(0 \lt n \lt
N\), partitions \(\mathbb{R}/\mathbb{Z}\) into component intervals
which each have one of at most 3 possible distinct lengths. Many
authors have explored higher dimensional generalizations of this
theorem. In this talk we will survey some of their results, and we
will explore a two-dimensional version of the problem, which turns out
to be closely related to the Littlewood conjecture. We will explain
how tools from homogeneous dynamics can be employed to obtain new
results about a problem of Erdős and Geelen and Simpson, proving the
existence of parameters for which the number of distinct gaps in a
higher dimensional version of the Steinhaus problem is unbounded. This
is joint work with Jens Marklof.
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