Dimension and Relative Frequencies

TL;DR

This paper provides an elementary method to compute the finite-state dimension of saturated sets of sequences with specified symbol frequency conditions, linking it to classical entropy and extending to effective dimensions.

Contribution

It offers a pointwise characterization of finite-state dimensions for saturated sets, showing their equivalence with Hausdorff and packing dimensions using elementary techniques.

Findings

Finite-state dimension equals Shannon entropy for saturated sets with specified frequencies.

Finite-state and strong dimensions coincide with Hausdorff and packing dimensions.

Results extend to computable and polynomial-time dimensions.

Abstract

We show how to calculate the finite-state dimension (equivalently, the finite-state compressibility) of a saturated sets $X$ consisting of {\em all} infinite sequences $S$ over a finite alphabet $\Sigma_m$ satisfying some given condition $P$ on the asymptotic frequencies with which various symbols from $\Sigma_m$ appear in $S$. When the condition $P$ completely specifies an empirical probability distribution $\pi$ over $\Sigma_m$, i.e., a limiting frequency of occurrence for {\em every} symbol in $\Sigma_m$, it has been known since 1949 that the Hausdorff dimension of $X$ is precisely $\CH(\pi)$, the Shannon entropy of $\pi$, and the finite-state dimension was proven to have this same value in 2001. The saturated sets were studied by Volkmann and Cajar decades ago. It got attention again only with the recent developments in multifractal analysis by Barreira, Saussol, Schmeling, and separately Olsen. However, the powerful methods they used -- ergodic theory and multifractal analysis -- do not yield a value for the finite-state (or even computable) dimension in an obvious manner. We give a pointwise characterization of finite-state dimensions of saturated sets. Simultaneously, we also show that their finite-state dimension and strong dimension coincide with their Hausdorff and packing dimension respectively, though the techniques we use are completely elementary. Our results automatically extend to less restrictive effective settings (e.g., constructive, computable, and polynomial-time dimensions).