The problem is to find limit for $\frac{1}{n}\sum_{k=1}^{[nx]} \sum_{j=1}^{n-1}k \sin \frac{\pi j}{n} \sin \frac{\pi k j}{n} \cos^{[tn^2]} \frac{\pi j}{n}$.
It is given that $x \in (0,1), t \in (0,1)$.
My ideas:
$$\frac{1}{n} \sum_{k=1}^{[nx]} k \sin \frac{\pi k j}{n} \sim n\int_0^x t \sin \left(\pi t j \right)dt,$$ when $j/N << 0$, then $$ \sin \frac{\pi j}{n} \cos^{[tn^2]} \frac{\pi j}{n} \sim \frac{\pi j}{n} \exp \left(-\frac{\pi^2 j^2 t}{2} \right),$$ so $$\frac{1}{n}\sum_{k=1}^{[nx]} \sum_{j=1}^{n-1}k \sin \frac{\pi j}{n} \sin \frac{\pi k j}{n} \cos^{[tn^2]} \frac{\pi j}{n} \rightarrow \pi \sum_{j=1}^\infty j \exp \left(-\frac{\pi^2 j^2 t}{2} \right) \int_0^x t \sin \left(\pi t j \right)dt,$$ but I don't know how to prove it mathematically.