For general Diophantine approximation, the Thue–Siegel–Roth theorem states that for any irrational algebraic number $x$, and any $\varepsilon>0$, there exists a constant $c=c(x,\varepsilon)$ such that $$ \left|x-\frac{p}q\right| > \frac{c}{q^{2+\varepsilon}}, $$ for any integers $p$ and $q>0$.

My first question is, is it possible to get larger lower bound, if we are allowed to choose $x$ carefully?

The second question is, does there exist $x$ such that $$ \left|x-\frac{p}{2^k}\right| > \frac{c}{2^{k}}, $$ for any $p$ and $k$, with some constant $c>0$?