Dimensions of Copeland–Erdös sequences

The base-k Copeland–Erdös sequence given by an infinite set A of positive integers is the infinite sequence CEk(A) formed by concatenating the base-k representations of the elements of A in numerical order. This paper concerns the following four quantities.•The finite-state dimension dimfs (CEk(A)), a finite-state version of classical Hausdorff dimension introduced in 2001.•The finite-state strong dimension Dimfs(CEk(A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of dimfs(CEk(A)) satisfying Dimfs(CEk(A)))⩾dimfs(CEk(A)).•The zeta-dimension (Dimζ(A), a kind of discrete fractal dimension discovered many times over the past few decades.•The lower zeta-dimension dimζ(A), a dual of Dimζ(A) satisfying dimζ(A)⩽Dimζ(A).We prove the following.dimfs(CEk(A))⩾dimζ(A). This extends the 1946 proof by Copeland and Erdös that the sequence (CEk(PRIMES)) is Borel normal.Dimfs(CEk(A))⩾Dimζ(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.