Let $$F=\{x \in R:||qx||\le2q^{1-\alpha}\log q \text{ for infinitely many } q \in \mathbb{R}\}$$ Show for $\alpha>2$, $\dim_H F\le 2/\alpha$.
Jarnik's theorem (By Falconer) says: Suppose $\alpha>2$, let $F$ be the set of real numbers $x \in [0,1]$ for which the inequality $||qx|| \le q^{1-\alpha}$ is satisfied by infinitely many positive integers q. Then $\dim_HF=2/\alpha$.
How to convert $||qx||\le2q^{1-\alpha}\log q$ into the inequality in the theorem so that it can be used for the proof?
Use $$ \log q\le q^{ε} $$ for arbitrary $ε>0$ and $q$ large enough (depending on $ε$).