The log moments of smallest denominators

TL;DR

This paper derives explicit formulas for the moments of the limit distribution of the smallest denominators of rationals in shrinking intervals with random centers, advancing understanding of minimal resonance orders in torus maps.

Contribution

It provides the first explicit formulas for the moments of the limit distribution in one dimension, answering open questions from prior numerical studies.

Findings

Explicit formulas for moments of the limit distribution in dimension one.

Convergence results for the logarithmic moments of smallest denominators.

Application to minimal resonance orders in torus maps.

Abstract

This paper studies the logarithmic moments of the smallest denominator of all rationals in a shrinking interval with random center. Convergence follows from the more general results in [arXiv:2310.11251, Bull. Lond. Math. Soc., to appear], and the key point of this note is the derivation of explicit formulas for the moments of the limit distribution in dimension one. This answers questions raised by Meiss and Sander in their numerical study of minimal resonance orders for torus maps with random rotation vectors [arXiv:2310.11600].