Quantitative equidistribution of angles of multipliers

We study angles of multipliers of repelling cycles for hyperbolic rational maps in $\mathbb{C}(z)$. For a fixed $K\gg 1$, we show that almost all intervals of length $2\pi \mathrm{/}K$ in $(-\pi ,\pi ]$ contain a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in $K$.