Product gales and Finite state dimension
Akhil S
TL;DR
This work develops a unified framework linking fractal dimension, automata theory, and entropy for sequences over finite alphabets. By introducing product gales and multi-bet finite-state gamblers, it proves that Hausdorff dimension can be characterised via $k$-product $s$-gales, and that the multi-bet finite-state dimension coincides with the classical finite-state dimension. It then establishes a tight connection between information-theoretic block-entropy rates and automata-based betting: the multi-bet FS-dimension equals the sliding block entropy rate, and sliding equals disjoint entropy rates, culminating in the equalities $H^{sliding}(X)=H^{disjoint}(X)= ext{dim}_{FS}(X)= ext{dim}_{FS}^{ ext{multi-bet}}(X)$. This provides a new automata-based proof pathway and a deeper understanding of how multiple notions of randomness and information density coincide for sequences on Cantor space.
Abstract
In this work, we introduce the notion of product gales, which is the modification of an $s$-gale such that $k$ separate bets can be placed at each symbol. The product of the bets placed are taken into the capital function of the product-gale. We show that Hausdorff dimension can be characterised using product gales. A $k$-bet finite-state gambler is one that can place $k$ separate bets at each symbol. We call the notion of finite-state dimension, characterized by product gales induced by $k$-bet finite-state gamblers, as multi-bet finite-state dimension. Bourke, Hitchcock and Vinodchandran gave an equivalent characterisation of finite state dimension by disjoint block entropy rates. We show that multi-bet finite state dimension can be characterised using sliding block entropy rates. Further, we show that multi-bet finite state dimension can also be charatcterised by disjoint block entropy rates. Hence we show that finite state dimension and multi-bet finite state dimension are the same notions, thereby giving a new characterisation of finite state dimension using $k$-bet finite state $s$-gales. We also provide a proof of equivalence between sliding and disjoint block entropy rates, providing an alternate, automata based proof of the result by Kozachinskiy, and Shen.