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The distribution of rationals in residue classes

Cristian Cobeli·Alexandru Zaharescu
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Abstract

Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity, the proportion of the $r$-tuples of consecutive denominators with components in certain apriori fixed arithmetic progressions with the same ratio approaches a limit, which is independent on the interval. The limit is given explicitly and it is completely described in a few particular instances.

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Cristian Cobeli, Alexandru Zaharescu. 2005-11-14. The distribution of rationals in residue classes. https://arxiv.org/abs/math/0511356

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