BCZ map is weakly mixing

Abstract

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.

Figures (3)

• Figure 1: Primitive Lattice points are marked as dots and others as crosses. If $x=1-b_k$ is the left vertical line and $x=1$ is the right vertical line, then for the illustrated lattice, the 4th flow return time to $\Lambda_0$ is the slope of the lower dotted line, while the 4th flow return time to $\Lambda_{b_k}$ is the slope of the upper dotted line. In particular, $R_k^{(4)}=5$ since the 4th return to $\Omega_k$ is the 5th return to $\Omega$.
• Figure 2: The $N_k$th slope lies in the shaded band with overwhelming probability; The box $B_1$ has expected number of primitive lattice points $\Omega(a)$, while the contribution to the expected number of lattice points in $B_2$ coming from lattices that contain at least two lattice points is $O(a^2)$.
• Figure 3: A typical set $A_{m,n}$. The height of the strip is $b_k/n$ and the $y$ axis intercept of the top line is $1/n$.

Theorems & Definitions (9)

• Theorem 1
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Lemma 3
• Claim 1
• Claim 2
• proof : Proof of Claim \ref{['claim:one']}
• proof : Proof of Claim \ref{['claim:two']}