Density of orbits of horocycle flows at sub-quadratic polynomial times
Adam Kanigowski, Maksym Radziwiłł
TL;DR
The paper analyzes the density and distribution of horocycle flow orbits on non-compact quotients at sparse times, proving density for times $\{n^{2-\delta}\}$ with $0<\delta<1$ and, conditionally on the Hardy–Littlewood conjecture, density for prime times. The key strategy blends approximating orbit segments by periodic orbits via a sheared transformation, Weyl-type equidistribution for polynomial phases, and a spectral theory framework for quadratic-time sampling. In the specific arithmetic setting $\Gamma=PSL(2,\mathbb{Z})$, it also proves equidistribution of quadratic-time sampled periodic orbits along primes congruent to $1\pmod{4}$ by reducing to subconvex bounds for $L$-functions twisted by quadratic characters. The results link dynamics of sparse sampling to deep number-theoretic inputs, including subconvexity bounds for automorphic L-functions and conjectures on prime patterns. Overall, the work advances understanding of how horocycle flows distribute when observed at sub-quadratic polynomial times and at prime times, revealing robust density and equidistribution phenomena in the non-compact setting.
Abstract
Let $Γ\subset PSL(2,\mathbb{R})$ be such that the space $X=Γ\backslash PSL(2,\mathbb{R})$ is not compact. Let $(h_t)$ be the horocycle flow acting on $X$. We show that for every $x\in X$ that is not periodic for $(h_t)$ and for every $δ\in (0,1)$ the orbit $\{h_{n^{2-δ}}x\}_{n\in \mathbb{N}}$ is dense in $X$. Assuming additionally the Hardy-Littlewood conjecture we show that for every non-periodic $x\in X$, $\{h_{p}x\}_{p- \text{prime}}$ is dense in $X$. Finally we show that for $Γ=PSL(2,\mathbb{Z})$, $\{h_{n^2}y_q\}_{n<q}$ equidistribute, as $q\to \infty$ along primes congruent to $1 \pmod{4}$, towards Haar measure, where $\{y_q\}$ is a sequence of periodic points of period $q$.