Normality in non-integer bases and polynomial time randomness

• Relationship between polynomial time randomness and normality in Pisot bases.
• Generalization of martingales to non-uniform distributions.
• Invariant Markov measures and martingales computed by a DFA.

It is known that if x∈[0,1]x∈[0,1] is polynomial time random then x is normal in any integer base greater than one. We show that if x   is polynomial time random and β>1β>1 is Pisot, then x is “normal in base β  ”, in the sense that the sequence (xβn)n∈N(xβn)n∈N is uniformly distributed modulo one. We work with the notion of P-martingale, a generalization of martingales to non-uniform distributions, and show that a sequence over a finite alphabet is distributed according to an irreducible, invariant Markov measure P if an only if no P-martingale whose betting factors are computed by a deterministic finite automaton succeeds on it. This is a generalization of Schnorr and Stimm's characterization of normal sequences in integer bases. Our results use tools and techniques from symbolic dynamics, together with automata theory and algorithmic randomness.