Abstract

By transferring to the world of functions computable by finite automata the classical theorem of numerical analysis establishing that every non-decreasing real valued function is almost everywhere differentiable, we obtain a characterization of the property of Borel normality. We consider functions mapping infinite sequences to infinite sequences and a notion of differentiability that, on the class of non-decreasing real valued functions, coincides with standard differentiability. We prove that the following are equivalent, for a real x in [0,1]:

(1) x is normal to base b.

(2) Every non-decreasing function computable by a finite automaton mapping infinite sequences to infinite sequences is differentiable at the expansion of x in base b.

(3) Every non-decreasing function computable by a finite automaton in base b mapping real numbers to real numbers is differentiable at x.

Joint work with Vernica Becher, Universidad de Buenos Aires.