Normality in non-integer bases and polynomial time randomness

TL;DR

This paper extends the concept of polynomial time randomness to non-integer bases, specifically Pisot bases, showing such randomness implies a form of normality, using automata theory and symbolic dynamics.

Contribution

It introduces $P$-martingales for non-uniform distributions and characterizes normality in non-integer bases via automata, generalizing previous integer base results.

Findings

Polynomial time randomness implies normality in Pisot bases.

$P$-martingales characterize distribution according to Markov measures.

Automata-based betting strategies determine sequence normality.

Abstract

It is known that if $x\in[0,1]$ is polynomial time random (i.e. no polynomial time computable martingale succeeds on the binary fractional expansion of $x$) then $x$ is normal in any integer base greater than one. We show that if $x$ is polynomial time random and $\beta>1$ is Pisot, then $x$ is "normal in base $\beta$", in the sense that the sequence $(x\beta^n)_{n\in\mathbb{N}}$ 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.