Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, denoted $T_\alpha\colon [0,1)\to[0,1)$ where $T_\alpha(x)=x+\alpha\bmod 1$. Let $C\subset [0,1)$ be a fat Canot set, that is a Cantor set with positive Lebesgue measure $\lambda(C)$. Set $\chi_C$ to be the characteristic function of $C$. What are the possible limit points of the Birkhoff averages of $\chi_C$ (a.k.a. Birkhoff spectrum of $\chi_C$), that is what can we say about the set $$ B(C)=\bigcup_{x\in[0,1)} S_C(x), $$ where $$ S_C(x)=\text{limit points of the sequence }\left(\frac1n\sum_{j=0}^{n-1}\chi_C(T^j_\alpha(x))\right)_{n=1}^\infty. $$ What can we say about the set $B(C)$?
Remarks. By the Birkhoff ergodic theorem for $\lambda$-a.e. $x\in [0,1)$ we have $$ \lim_{n\to\infty} \frac1n\sum_{j=0}^{n-1}\chi_C(T^j_\alpha(x))=\lambda(C). $$ Furthermore, by the unique ergodicity of $T_\alpha$ we have $$ \limsup_{n\to\infty} \frac1n\sum_{j=0}^{n-1}\chi_C(T^j(x))\le\lambda(C). $$ But $C$ is nowhere dense, so $U=[0,1)\setminus C$ is nonempty, open and dense, and the same holds true for $T_\alpha^{-k}(U)$ for $k=1,2,\ldots$. Therefore the set $$ D=\bigcap_{k=0}^\infty T_\alpha^{-k}(U) $$ is residual. For $x\in D$ we clearly have $S_C(x)=\{0\}$. It follows that $0,\lambda(C)\in B(C)$ and $B(C)\subseteq [0,\lambda(C)]$. Can it happen that $B(C)=\{0,\lambda(C)\}$? Can it happen that $B(C)= [0,\lambda(C)]$?
