The distribution of rationals in residue classes
Cristian Cobeli, Alexandru Zaharescu
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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