Smallest denominators

Abstract:

If we partition the unit interval into 3000 equal subintervals and take the smallest denominator amongst all rational points in each subinterval, what can we say about the distribution of those 3000 denominators? I will discuss this and related questions, its connection with Farey statistics and random lattices. In particular, I will report on higher dimensional versions of a recent proof of the 1977 Kruyswijk-Meijer conjecture by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305] on the convergence of the expectation value of the above distribution, as well as closely related work by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405--1413]. In fact, we will uncover the full distribution and prove convergence of more moments than just the expectation value. (This I believe was previously not known even in one dimension.) 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.