Dimensions of Copeland-Erdös Sequences

Abstract

The base-k Copeland-Erdös 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 finite-state dimension dim FS (CE k (A)), a finite-state version of classical Hausdorff dimension introduced in 2001.• The finite-state strong dimension Dim FS (CE k (A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of dim• The zeta-dimension Dim ζ (A), a kind of discrete fractal dimension discovered many times over the past few decades.• The lower zeta-dimension dim ζ (A), a dual of Dim ζ (A) satisfying dim ζ (A) ≤ Dim ζ (A).We prove the following.. This extends the 1946 proof by Copeland and Erdös that the sequence CE k (PRIMES) is Borel normal. Dim FS (CE3. 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.