Sums of reciprocals and the three distance theorem

In this paper we investigate the sums of reciprocal to an arithmetic progression mod 1. Bounds for these sums have been studied for a long while. In this paper we develop an alternative technique using the so-call three distance theorem which allows us to recover some of the known results in a more efficient form and obtain new bounds

Polygons in billiard orbits

We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these polygons.Comment: 14 page

Symmetries of the Three Gap Theorem

The Three Gap Theorem states that for any $\alpha \in \mathbb{R}$ and $N \in \mathbb{N}$, the fractional parts of $\{ 0\alpha, 1\alpha, \dots, (N - 1)\alpha \}$ partition the unit circle into gaps of at most three distinct lengths. We prove a result about symmetries in the order with which the sizes of gaps appear on the circle.Comment: To appear in The American Math Monthly. This is the author's original manuscript (AOM), and this version does not include several improvements that took place due to the refereeing process. Comments welcome

A concise geometric proof of the three distance theorem

The three distance theorem states that for any given irrational number $\alpha$ and a natural number $n$, when the interval $( 0, 1 )$ is divided into $n+1$ subintervals by integer multiples of $\alpha$, namely, $\{0\}, \{ \alpha \}, \{ 2\,\alpha \},\dots, \{ n\,\alpha \}$, then each subinterval is limited to at most three different lengths. Steinhaus conjectured this theorem in the 1950s, and many researchers have given various proofs since then. This paper aims to improve the perspective by showing a two-dimensional map of how the unit interval is divided by continuously changing $\alpha$, and give a concise proof of the theorem. We also present a simple proof of the three gap theorem, a dual of the three distance theorem.Comment: Slightly modified the English wording of Abstract. Changed the notation of fractional parts and intervals to the common ones. Corrected the notation of reference