Let $a_{N,k}$ be non-negative numbers for $N,k = 0, 1, \ldots$, satisfying $$ \sum_{k=0}^\infty a_{N,k} =1 $$ for every $N$.

Let $T$ be the rotation of irrational angle $\gamma$. Define the weighted averages $$ S_N(f) = \sum_{k=0}^\infty a_{N,k} f \circ T^k. $$ Say that the weights $a_{N,k}$ are good if $S_N(f) \overset{L^2}{\longrightarrow} \int f$ for every $f \in L^2$.

The following result is known: if the weights are good, then $$ (\star)\colon \quad \lim_{N \to \infty} \sum_{\substack{k=0 \\ k \in Q_{b}}}^\infty a_{N,k} = b $$ for every $b \in (0,1)$, where $$ Q_{b} = \bigl\{k \in \mathbb{N} \mid \{k\gamma\} \leq b \bigr\}. $$

Conversely, is it true that $(\star)$ implies that the weights are good ?


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