I am interested in the Lebesgue measure of the set $$S_{N,\varepsilon}:=\{x\in[0,1]:\min_{1\leq n\leq N}|\sin(\pi n x)|<\varepsilon\}.$$ I know this is basically asking if $x$ can be well approximated by a rational. So I can recast this as finding the Lebesgue measure of the set
$$T_{N,\varepsilon}:=\{x\in[0,1]:\min_{1\leq n\leq N}\{nx\}<\varepsilon\},$$
where $\{\cdot\}$ denotes the fractional part of $\cdot$. As $N$ increases, obviously the Lebesgue measure goes to $1$. But as a function of $N$, what is it?
