The Minimal Denominator Function and Geometric Generalizations

TL;DR

This paper offers a geometric interpretation of the minimal denominator function, computes its limiting distribution, and extends the concept to higher-dimensional settings using equidistribution in Lie group actions.

Contribution

It introduces a geometric perspective for the minimal denominator function and generalizes it to higher dimensions via unipotent orbit equidistribution.

Findings

Derived the limiting distribution of the normalized minimal denominator function.

Extended the concept to higher-dimensional lattices and translation surfaces.

Connected the problems to equidistribution of unipotent orbits in Lie groups.

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.