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This paper investigates the set of angles of the parameter rays which land onthe real slice $[-2,1/4]$ of the Mandelbrot set. We prove that this set haszero length but Hausdorff dimension 1. We obtain the corresponding results forthe tuned images of the real slice. Applications of these estimates in thestudy of critically non-recurrent real quadratics as well as biaccessiblepoints of quadratic Julia sets are given.

External rays and the real slice of the Mandelbrot set