Finite-State Dimension and Real Arithmetic

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.Comment: 15 page

Dimensions of Copeland-Erdos Sequences

The base-$k$ {\em Copeland-Erd\"os sequence} given by an infinite set $A$ of positive integers is the infinite sequence \CE_k(A) formed by concatenating the base-$k$ representations of the elements of $A$ in numerical order. This paper concerns the following four quantities. The {\em finite-state dimension} \dimfs (\CE_k(A)), a finite-state version of classical Hausdorff dimension introduced in 2001. The {\em finite-state strong dimension} \Dimfs(\CE_k(A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of \dimfs(\CE_k(A)) satisfying \Dimfs(\CE_k(A)) \geq \dimfs(\CE_k(A)). The {\em zeta-dimension} \Dimzeta(A), a kind of discrete fractal dimension discovered many times over the past few decades. The {\em lower zeta-dimension} \dimzeta(A), a dual of \Dimzeta(A) satisfying \dimzeta(A)\leq \Dimzeta(A). We prove the following. \dimfs(\CE_k(A))\geq \dimzeta(A). This extends the 1946 proof by Copeland and Erd\"os that the sequence \CE_k(\mathrm{PRIMES}) is Borel normal. \Dimfs(\CE_k(A))\geq \Dimzeta(A). These bounds are tight in the strong sense that these four quantities can have (simultaneously) any four values in $[0,1]$ satisfying the four above-mentioned inequalities.Comment: 19 page

Normal numbers and finite automata

We give an elementary and direct proof of the following theorem: A real number is normal to a given integer base if, and only if, its expansion in that base is incompressible by lossless finite-state compressors (these are finite automata augmented with an output transition function such that the automata input–output behaviour is injective; they are also known as injective finite-state transducers). As a corollary we obtain V.N. Agafonov’s theorem on the preservation of normality on subsequences selected by finite automata.Fil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Heiber, Pablo Ariel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Normal numbers and finite automata

We give an elementary and direct proof of the following theorem: A real number is normal to a given integer base if, and only if, its expansion in that base is incompressible by lossless finite-state compressors (these are finite automata augmented with an output transition function such that the automata input–output behaviour is injective; they are also known as injective finite-state transducers). As a corollary we obtain V.N. Agafonov’s theorem on the preservation of normality on subsequences selected by finite automata.Fil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Heiber, Pablo Ariel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin