Smallest denominators

Abstract

We establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405-1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk-Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one-dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130-149] as well as new results on pigeonhole statistics.

Figures (3)

• Figure 1: The limit density $\eta(s)$ compared to the distribution of the smallest denominator of rationals in each interval $[\frac{j}{3000},\frac{j+1}{3000})$, $j=0,\ldots,2999$, cf. Section \ref{['secDiscrete']}. The same law describes the shortest cycle length of a large random circulant directed graph of (in- and out-) degree 2 circulant.
• Figure 2: The function $M(\alpha)$, with the height of the graph representing its absolute value and the colour its argument.
• Figure 3: The limiting moments $M(\alpha)$ for real $\alpha$ (blue) compared with finite-$N$ approximations (red) corresponding to the left hand side of \ref{['LD1Mom22']} with $N=100$, $50$, $25$ (top to bottom) and $n=1$, $\alpha=1$, ${\mathcal{D}}=[0,1)$, ${\mathcal{A}}=[0,1)$, ${\text{\boldmath$x$}}_0=0$.

Theorems & Definitions (21)

• Proposition 1
• proof
• Proposition 2
• Proposition 3
• proof
• Proposition 4
• proof : Proof when $\operatorname{Re}\alpha=0$.
• proof : Proof when $\operatorname{Re}\alpha>0$.
• proof : Proof when $\operatorname{Re}\alpha<0$.
• Proposition 5