BCZ map is weakly mixing

Yitwah Cheung, Anthony Quas

Published 2024-03-22Version 1

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.