Dimensions of Copeland-Erdos Sequences

Abstract

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.