We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Prove the Ergodic Theorem for the function $f$. (One may set $M = 2 \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \|$. For given $\epsilon > 0$, one may choose $k \in \mathbb { N } ^ { * }$ such that $d _ { k } ( \epsilon ) < \epsilon$. Next, one may apply Question 14 to $f _ { k }$ and consider $T _ { 0 } > 0$ such that for every $T \geq T _ { 0 }$: $$\left| \frac { 1 } { T } \int _ { 0 } ^ { T } f _ { k } \circ \theta ( t ) d t - ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f _ { k } \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \right| < \epsilon .)$$