Expression of closed intervals appeared in the $m$-th stage of construction of the Cantor set
No, since the interval $[1, 3^{-m}]$ appears in the $m-$th stage of the construction of the Cantor set, however, $\frac{n + 1/3}{3^m} \ne 1$ for any $n, m$ since otherwise we have $$ 3n + 1 = 3^{m+1} $$ which is impossible since $3$ does not divide $1$.