Finite-State Dimension and Real Arithmetic

TL;DR

This paper proves that for any base, rational, and real number, certain expansions share the same finite-state dimensions, extending Wall's 1949 theorem on Borel normality under addition and multiplication.

Contribution

It introduces a new proof using entropy rates and Schur concavity to establish invariance of finite-state dimensions under rational operations.

Findings

Finite-state dimensions are invariant under addition and multiplication by nonzero rationals.

Extends Wall's 1949 theorem on Borel normality.

Provides a new proof technique for properties of number expansions.

Abstract

We use entropy rates and Schur concavity to prove that, for every integer k >= 2, every nonzero rational number q, and every real number alpha, the base-k expansions of alpha, q+alpha, and q*alpha all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.