On simply normal numbers with digit dependencies

Abstract

Given an integer (Figure presented.) and a set (Figure presented.) of prime numbers, the set (Figure presented.) of Toeplitz numbers comprises all elements of [0, b[whose digits (Figure presented.) in the base-b expansion satisfy (Figure presented.) for all (Figure presented.) and (Figure presented.). Using a completely additive arithmetical function, we construct a number in (Figure presented.) that is simply Borel normal if, and only if, (Figure presented.). We then provide an effective bound for the discrepancy.