The orbit in a two dimensional special region as viewed from a lattice in one dimension higher

This picture is closely related to the previous one. Here the green parallelogram, considered as a subset of \(\mathbb{R}^2\), is a bounded remainder set for an irrational rotation by \(\alpha=(\alpha_1,\alpha_2)\in\mathbb{T}^2\) (see the description of the previous image for more information about bounded remainder sets). For the purposes of this discussion let us call this set \(A\).

The red and blue dots represent points of the lattice \(\Lambda\subseteq\mathbb{R}^3\) generated by the vectors \[\left(\begin{array}{c}\alpha_1\\\alpha_2\\ 1\\\end{array}\right),\left(\begin{array}{c}1\\0\\ 0\\\end{array}\right),~\text{and}~\left(\begin{array}{c}0\\1\\ 0\\\end{array}\right).\] The region \(A\) is the projection of the parallelogram generated by two vectors \(v_1,v_2\in\Lambda\), and these vectors together with a third vector \(v_3\) (which in the picture is the vector from the bottom blue dot to the one next one up) form a basis for the lattice. The vectors \(v_1\) and \(v_2\) span a plane in \(\mathbb{R}^3\), which we denote by \(H_0\), and for each \(k\in\mathbb{Z}\) we define \(H_k=H_0+kv_3.\) The lattice \(\Lambda\) is the disjoint union of the sets \(H_k\cap\Lambda\), and parts of three of these sets (corresponding to \(k=1,2,\) and \(3\) are shown in the picture.

To link this picture with return times to \(A\) of rotation by \(\alpha\) in \(\mathbb{T}^2\), notice that \(\Lambda\) consists of all vectors of the form \[\left(\begin{array}{c}n\alpha_1-a_1\\n\alpha_2-a_2\\ n\\\end{array}\right),\] where \(n, a_1\), and \(a_2\) are integers. The first two coordinates of such a vector keep track of the position in \(\mathbb{R}^2\) of the orbit of \(0\) under rotation by \(\alpha\), while the third coordinate keeps track of the number of rotations which have been applied to reach the given position. Therefore we can study the return times of \(0\) to \(A\) by looking at the collection of all points in \(\Lambda\) which lie in the cylinder above \(A\) in \(\mathbb{R}^3\), and taking these points in order from the base upwards towards infinity corresponds to looking at all returns to \(A\) in the order in which they occur. The overlying lattice structure is what enables us to prove that sets like \(A\) are bounded remainder sets. Detailed proofs are available in a joint paper with Henna Koivusalo, available at arxiv:1402.2125.

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