Elements of information theory
Elements of information theory
Note: fractal dimension and logarithmic loss unpredictability
Theoretical Computer Science
Dimension in Complexity Classes
SIAM Journal on Computing
Theoretical Computer Science
Entropy rates and finite-state dimension
Theoretical Computer Science
Effective Strong Dimension in Algorithmic Information and Computational Complexity
SIAM Journal on Computing
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
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The base-kCopeland-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.