Given an integer π>0 and a set P of prime numbers, the set T_P of Toeplitz numbers comprises all elements of [0,π) whose digits (π_π)_{n>0} in the base-π expansion satisfy π_π = π_ππ forall π in P and π>0. Using a completely additive arithmetical function, we construct a number in T_P that is simply Borel normal if, and only if the sum of 1/π over all π in P diverges; we provide an effective bound for the discrepancy. For finite P we show that almost every number in T_P is Borel normal to base π. For P={2} we show more: almost every number in T_P is Borel normal to all integer bases.
Joint work partly with Christoph Aistleitner and Olivier Carton, and partly with Agustín Marchionna and Gérald Tenenbaum.