  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.
