Mathematical analysis reveals density and calculates limiting distributions of scaled denominators in consecutive Farey sequences, highlighting statistical regularities in number theory.
The sequence [Formula: see text] of [Formula: see text]-saturated Farey fractions was defined in our previous work by [Formula: see text], where [Formula: see text] is the multiplicative inverse of [Formula: see text] in [Formula: see text]. Here, we prove that the set of [Formula: see text]-scaled denominators of consecutive fractions in [Formula: see text] is dense in the region [Formula: see text], and provide a formula for their distribution in [Formula: see text] as [Formula: see text].
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Anderson et al. (2026) studied this question.
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