1. Give an irrational $x$ (We can limit it to $[0,1]$). Prove that there exists infinitely many fractions $\dfrac pq$ satisfying that $\gcd(p,q)=1$ such that $$\left|x-\frac pq\right|\le\frac1{q^2}.$$ Furthermore, prove that the set of those $x \in \mathbb R$ such that there exist infinitely many fraction $\dfrac pq$ satisfying that $\gcd(p,q)=1$ such that $$\left|x-\frac pq\right|\le \frac1{q^{2+ε}}\qquad ε>0$$ is a set of measure zero.
This question seems to be basically important since it indicates the speed of approximation to a real number by rationals.And I think it can use to solve the following question.
2. My teacher told in class that Riemann function R(x) is not derivable in irrational point and neither square of Riemann function $R^2\left(x\right)$,while $R^3\left(x\right)$ is derivable at x= $\sqrt{\frac{p}{q}}$,which cannot be extracted.Furthermore,$R^n(x)$ is derivable at x=$\sqrt[n-1]{\frac{p}{q}}$.How to prove that?