Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games

Question

A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have

$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$

The set of badly approximable is known to have Hausdorff dimension one (full). The classical proof was due to W. Schmidt where he introduced the famous Schmidt games and showed that the set of badly approximable numbers are "winning" in his game and any winning set has Hausdorff dimensions (elementary but a bit lengthy; further, Schmidt's tool works for generalizations of it in higher dimensions).

Are there any proofs for badly approximable numbers having full Hausdorff dimension without using Schmidt games? Elementary proofs are welcome. But please also feel free to use Dani's Correspondence principle (badly approximable numbers correspond to "bounded trajectories") or continued fractions if inevitable. Any references are welcome!

Answer 1

For $x \in \mathbb{R}$ and $r \in \mathbb{N}$, let $K_r(x)$ be the minimal Kolmogorov complexity of a rational number in $]x - 2^{-r}, x + 2^{-r}[$. The effective Hausdorff dimension of $x$ then is defined as: $$\mathrm{dim}(x) = \liminf_{r \to \infty} \frac{K(r)}{r}$$

We can relativize the effective Hausdorff dimension to some oracle $p \in 2^{\omega}$, written as $\mathrm{dim}(x)^p$. This just means that the Kolmogorov complexity is measured relative to $p$. Now the point-to-set-principle by Lutz and Lutz 1 tells us how the relativized effective Hausdorff dimension and the usual Hausdorff dimension of a set are related: $$\mathrm{dim}_\mathcal{H}(A) = \min_{p \in 2^\omega} \sup_{x \in A} \mathrm{dim}^p(x)$$

In particular we see that having high Hausdorff dimension is all about having "badly approximable" elements in a more general sense!

Thus, to see that the set of badly approximable real numbers has Hausdorff dimension $1$, it suffices to construct for any $q < 1$ and any oracle $p \in 2^{\omega}$ a badly approximable real with $\mathrm{dim}^p(x) \geq q$. This will be rather straight-forward: To figure out what $x$ should be up to precision $2^{-r}$, we just need to avoid whatever approximations programs of length $\leq r$ would output, as well as points of the form $p/2^{-r/2}$.

1 Lutz & Lutz : "Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension", ACM Transactions on Computation Theory: Link