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
