Dimensions of Copeland-Erdoes sequences

摘要

The base-k Copeland-Erdoes 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. rn1. The finite-state dimension dim_(FS)(CE_k(A)), a finite-state version of classical Hausdorff dimension introduced in 2001. rn2. 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_(FS)(CE_k(A)) satisfying Dim_(FS)(CE_k(A)) ≥ dim_(FS)(CE_k(A)). rn3. The zeta-dimension Dim_ζ(A), a kind of discrete fractal dimension discovered many times over the past few decades. rn4. The lower zeta-dimension dimζ(A), a dual of Dim_ζ(A) satisfying dim_ζ (A) ≥ Dim_ζ(A). rnWe prove the following. rn(1) dim_(FS)(CE_k(A)) ≥ dim_ζ(A). This extends the 1946 proof by Copeland and Erdos that the sequence CE_k(PRIMES) is Borel normal. rn(2) Dim_(FS)(CE_k(A)) ≥ Dim_ζ(A). rn(3) 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.