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.

中文翻译：

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最小分母

我们建立了 Chen 和 Haynes 最近定理的高维版本 [Int. J. Number Theory 19 (2023), 1405–1413] 关于小长度随机移位区间内有理点最小分母的期望值，以及最近由 Balazard 和 Martin 证明的密切相关的 1977 Kruyswijk-Meijer 猜想 [公牛。科学。数学。187（2023），论文编号103305]。我们用多维法雷分数的空统计来表达最小分母的分布，并证明了分布函数和某些有限矩的收敛性。即使在一维环境中，后者以前也是未知的。我们还获得了 Kargaev 和 Zigljavsky 关于 Farey 序列距离函数矩的工作的更高维扩展 [J. Number Theory 65 (1997), 130–149] 以及鸽笼统计的新结果。