Egoroff’s theorem and maximal run length

We construct a sequence of measurable functions converging at each point of the unit interval, but the set of points with any given rate of convergence has Hausdorff dimension one. This is used to show that a version of Egoroff’s theorem due to Taylor is best possible. The construction relies on an analysis of the maximal run length of ones in the dyadic expansion of real numbers. It is also proved that the exceptional set for a limit theorem of Renyi has Hausdorff dimension one.