The Sato–Tate independence conjecture for successive Frobenius angles

Let $E$ be an elliptic curve over $\mathbf{Q}$ without complex multiplication, and let $x_k\in[0,1]$ be its normalized Frobenius angle at the $k$-th prime. For each integer $s\geq1$, define

$$X_k=(x_k,x_{k+1},\ldots,x_{k+s-1})\in[0,1]^s.$$

Let $\mu_{ST}^{[s]}$ be the product of $s$ copies of the Sato–Tate measure $\mu_{ST}$.

The Sato–Tate independence conjecture. For every integer $s\geq1$, the sequence $\{X_k\}_{k\geq1}$ is uniformly distributed with respect to $\mu_{ST}^{[s]}$; equivalently,

$$\frac{1}{K}\sum_{k=1}^{K}\delta_{X_k}$$

converges weakly to $\mu_{ST}^{[s]}$ as $K$ tends to infinity. The case $s=1$ is the solved Sato–Tate theorem, while the cases $s\geq2$ assert statistical independence of successive Frobenius angles and remain open.