Normality and Differentiability
Seminar Room 1, Newton Institute
AbstractBy 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 Verónica Becher, Universidad de Buenos Aires.