The Minimal Denominator Function and Geometric Generalizations

Abstract

We provide a geometric interpretation for a normalized version of the minimal denominator function, $$q_{\min }(x,\delta )=\min \left\{q\in \mathbb{N}:\text{ there exists }p\in \mathbb{Z}\text{ such that }\frac{p}{q}\in (x-\delta ,x+\delta )\right\},$$ introduced by Chen and Haynes. We use this interpretation to compute the limiting distribution of a suitably normalized version of $q_{\min }(x,\delta )$ as a function of $x$, and give generalizations of the idea of minimal denominators to higher-dimensional unimodular lattices, linear forms, and translation surfaces. The key idea is to turn this circle of problems into equidistribution problems for translates of unipotent orbits of a Lie group action on an appropriate moduli space.

16 pages, 6 figures