almost periodic visits of a compact set of a circle group rotation

Question

Let $\mathbb{T}$ be the circle group and $R:\mathbb{T\rightarrow T}$ an irrational rotation (hence a minimal system).

Suppose there exists a compact set $K\subset\mathbb{T}$ such that for every $x\in\mathbb{T}$ there exists $n\geq0$ such that $R^{n}(x)\in K.$

Does this imply every point enters $K$ almost periodically i.e. there exists $p>0$ such that for every $x$ there exists $0\leq t\leq p$ such that $R^{t}(x)\in K?$

Answer 1

If $K$ has empty interior then $R^{-j}(K)$ does also. Then by the Baire category theorem $\cup_{j=0}^{\infty}R^{-j}K$ is not the whole set. Therefore $K$ has non-empty interior, and ... I assume the argument is easily finished in this case.