Let's call $\gamma \in \mathbb R$ $\alpha$-badly approximable if there is some $d>0$ such that for all pairs $(k,l)\in \mathbb Z^2$ such that $ l\neq 0,$
$$ \left| \gamma - \frac kl \right| > \frac d{|l|^\alpha} .$$
I was simply wondering the following:
Question 1. Take for example $\displaystyle \gamma = \frac{\log(3)}{\log(2)}$. Is there any known value of $\alpha$ for which $\gamma$ is $\alpha$-badly approximable?
Furthermore, it is well-known that a number is 2-badly approximable (a.k.a. Diophantine) if and only if the sequence of integers corresponding to its continued fraction expansion is bounded.
For example, the golden ratio $\phi = [1;1,1,1,\ldots]$ and $\sqrt 2 = [1;2,2,2,\ldots]$ (or any non-rational algebraic number).
Question 2. Can one deduce any properties of the continued fraction expansion if a number is $\alpha$-badly approximable in general, or indeed deduce $\alpha$-badly approximability from the continued fraction expansion?
Question 3. Is it at all easy to provide for a given $\alpha > 2$, examples of numbers which are $\alpha$-badly approximable but not Diophantine?
(For any such $\alpha>2$ this is Lebesgue almost-always true by Jarnik's Theorem.)