The log moments of smallest denominators

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].