This picture illustrates the induced dynamics in a bounded remainder set for an irrational rotation of \(\mathbb{T}^2:=\mathbb{R}^2/\mathbb{Z}^2\). For those who are not familiar with the terminology, given \(\alpha\in\mathbb{R}^s\), a measurable set \(A\subseteq\mathbb{T}^s\) is a called a bounded remainder set for \(\alpha\) if \[\sup_{x\in\mathbb{T}^s} \sup_{N\in\mathbb{N}}\left|\sum_{n=1}^N\chi_A (x+n\alpha )-N|A|\right| <\infty , \] where \(|A|\) denotes the Lebesgue measure of \(A\). In other words, \(A\) is a bounded remainder set if the time and space averages of rotation by \(\alpha\) always differ by at most some universal constant which is independent of the initial point.

Bounded remainder sets are relatively uncommon, but they have important applications in several areas of mathematics. It is known that, in any dimension, there are no `non-trivial' examples of aligned boxes (boxes with all faces parallel to coordinate planes) which are bounded remainder sets. This was proved in a 1987 paper by Liardet, which is available from the European Digital Mathematics Library by following this link. In his paper, Liardet also developed a method for using non-trivial bounded remainder set in some dimension to construct bounded remainder sets in one dimension higher.

Recently, in joint work with Henna Koivusalo (available at arxiv:1402.2125), we found a simple algorithm for generating bounded remainder sets for any irrational rotation in any dimension. Our sets are parallelotopes which are obtained as projections of special fundamental domains of a lattice in one dimension higher. In the picture above, the bounded remainder set \(A\) is the parallelogram bounded by the solid black lines. The blue dot which is furthest left corresponds to the point \(0\in\mathbb{T}^2\), and the other blue dots are the successive return times of a particular irrational rotation to \(A\).

Notice that the outer boundary of the diagram is a hexagon, consisting of two solid and four dashed lines. This hexagon is the projection of a fundamental domain of a lattice in \(\mathbb{R}^3\). The blue dots move along straight lines, corresponding to the projection of one of the edge vectors of the fundamental domain, until they exit the region \(A\) (as illustrated by the red dots). They are then translated back into \(A\) by a vector corresponding to one of the other edges of the fundamental domain. This characterization of the dynamics leads to a proof that such sets are bounded remainder sets.

One consequence of our theorems about bounded remainder sets is that we are able to explain how to construct a large collection of quasi-crystals (aperiodic mathematical structures which occur naturally in the world around us) which are bounded distance to lattices.