External rays and the real slice of the Mandelbrot set

This paper investigates the set of angles of the parameter rays which land on the real slice $[-2,1\mathrm{/}4]$ of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for the tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as biaccessible points of quadratic Julia sets are given.