  The BCZ map was introduced in 2001 by Boca, Cobeli and Zaharescu as a tool to
study the statistical properties of Farey sequences, whose relation to Riemann
Hypothesis dates back to Franel and Landau. Later, J. Athreya and the first
author observed that the BCZ map arises as a Poincare section of horocycle
flow, establishing both ergodicity as well as zero measure-theoretic entropy.
In this article, we prove that the BCZ map is weakly mixing, answering the last
remaining question about the BCZ map raised in a 2006 survey by Boca and
Zaharescu. The proof uses a self-similarity property of the BCZ map that
derives from a well-known fact that horocycle flow is renormalized by the
geodesic flow, a property already observed in arXiv:1206.6597. We note that the
questions of mixing and rigidity remain open.
