Measure of the set of real numbers that can be approximated in this way

Question

Let $$A = \{x \in \mathbb{R}\mid \exists\,\text{infinitely many pairs of integers $p,q$ such that $|x-p/q| \leq 1/q^3$}\}.$$ Is the measure of $A$ equal to $0$? Any ideas?

Answer 1

Let's restrict to $(0,1]$ and let $A_{q}=\bigcup_{p=1}^q B(\frac pq,\frac 1{q^3})$. Then $\mu(A_{p,q})\le \frac 1{q^2}$. We are looking for $$ \mu(A\cap(0,1])\le \mu\left(\bigcap_{n=1}^\infty\bigcup_{q=n}^\infty A_q\right)\le \mu\left(\bigcup_{q=n}^\infty A_q\right)\le \sum_{q=n}^\infty\mu(A_q)\le\sum_{q=n}^\infty\frac1{q^2}.$$ Since the last expression is the tail of the convergent series $\sum_{q=1}^\infty\frac1{q^2}(=\frac{\pi^2}{6})$, it tends $\to0$ as $n\to\infty$. We conclude $\mu(A\cap(0,1])=0$ and as $A$ is the countable union of translated copies of $A\cap(0,1]$, we also have $\mu(A)=0$.