Lebesgue measure of the set of ratios $x/y$ with $x,y$ in a fat Cantor set

Question

Suppose $I_\alpha\subset[0,1]$ is an $\alpha$-Cantor set of Lebesgue measure $\alpha$ and let $I_\alpha^*=I_\alpha\setminus\{0\}$. What is the Lebesgue measure of the following set $$\{\frac{x}{y}:x,y\in I_\alpha^*\}?$$

Answer 1

Let $E$ be any measurable set. Then for $t>0,$ $m(tE) = tm(E).$

Suppose now $E \subset [0,1]$ has positive measure and $E$ contains a positive sequence $y_n \to 0^+.$ Then $m((1/y_n)E) = (1/y_n)m(E) \to \infty.$ Since $R=\{x/y: x,y\in E, x,y\ne 0\}$ contains $(1/y_n)E\setminus \{0\}$ for every $n,$ we have $m(R)=\infty.$

Now your $I_\alpha $ contains a sequence $y_n \to 0^+.$ Thus your set of ratios has measure $\infty;$ the fact that $I_\alpha$ is a Cantor set doesn't really matter.