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.