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Intervals Between Farey Fractions in the Limit of Infinite Level

Jan Fiala·Peter Kleban
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Abstract

The modified Farey sequence consists, at each level k, of rational fractions r_k^{(n)}, with n=1, 2, ...,2^k+1. We consider I_k^{(e)}, the total length of (one set of) alternate intervals between Farey fractions that are new (i.e., appear for the first time) at level k, I^{(e)}_k := \sum_{i=1}^{2^{k-2}} (r_k^{(4i)}- r_k^{(4i-2)}) . We prove that \liminf_{k\to \infty} I_k^{(e)}=0, and conjecture that in fact \lim_{k \to \infty}I_k^{(e)}=0. This simple geometrical property of the Farey fractions turns out to be surprisingly subtle, with no apparent simple interpretation. The conjecture is equivalent to $ lim_{k \to \infty}S_{k}=0, where S_{k} is the sum over the inverse squares of the new denominators at level k, S_{k}:=\sum_{n=1}^{2^{k-1}} 1/ (d_k^{(2n)} )^2. Our result makes use of bounds for Farey fraction intervals in terms of their "parent" intervals at lower levels.

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Jan Fiala, Peter Kleban. 2009-12-02. Intervals Between Farey Fractions in the Limit of Infinite Level. https://arxiv.org/abs/math-ph/0505053

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