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arXiv · 2308.08076

The Minimal Denominator Function and Geometric Generalizations

Albert Artiles
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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.

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Albert Artiles. 2023-08-16. The Minimal Denominator Function and Geometric Generalizations. https://arxiv.org/abs/2308.08076

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