Extractable patterns

In this post we attempt to understand a relationship between Hausdorff dimension defined in a classical way and Hausdorff dimension defined using gales. Recall that classical Hausdorff dimension is defined in terms of covers. So, in order to understant that relationship, we need to be able to 'switch' back and forth from covers and gales. In this post we will try to switch from covers to gales. As before, we work in Cantor space $\mathcal{C} = \{0,1\}^{\mathbb{N}}$. We start with some subset $X\subseteq \mathcal{C}$. For any given $\varepsilon>0$, one can choose $s\ge 0$ such that $s-\varepsilon < \dim_H(X) < s$. Then we go on to show that there $s$-gale succeeding on $X$, i.e. $\inf G(X)\le s$. Combining this with above inequality, we have \[ \inf G(X)\le s < \dim_H(X) + \varepsilon \] By letting $\varepsilon \to 0$, one shows that $\inf G(X)\le \dim_H(X)$. So, we are left to show that it is indeed possible to construct $s$-gale given the fact that $\dim...

In this post we aim to provide characterization of Hausdorff Dimension in Cantor space in terms of special function called gales. Our presentation mainly follows that of Mayordomo [1], while the notion itself is from Lutz [2]. Gales Let us first go through definitions of supergales and gales. Take $s\in [0, \infty)$. An $s$-supergale is a function $d:\{0,1\}^*\to [0,\infty)$ such that: \[ d(w) \ge 2^{-s}\left[ d(w0) + d(w1) \right] \] for all $w\in \{0,1\}^*$. A definition $s$-gale is similar to the above, except for the equality sign in place of the inequality sign for each $w\in \{0,1\}^*$. A supermartingale is $1$-supergale, while a martingale is an $1$-gale. As for names, they seem to derive from a notion of martingale from probability theory. A parameter $s$ describes fairness of the betting game. Observe that the condition for $s$-gale can be rewritten as: \[ E[d(wb)] = \frac{1}{2}d(w0) + \frac{1}{2}d(w1) = 2^{s-1}d(w) \] Depending on the value of $s$, the game ca...