Suppose, we are given sequence $x_k= k\mod 2\pi$ and sequence $y_{k,n}$, where $y_{k,n},0\leq n\leq k$ is just sequence $x_0,\dots,x_k$ sorted in increasing order for a given integer $k$. What can we say about the sum $\sum \limits_{n=1}^{k}\big( \sum\limits_{t=0}^{n}\frac{2\pi}{k}-(y_{k,t+1}-y_{k,t}) \big)$? Is it bounded? Does it have a limit when $k \to \infty$?
I know that $x_k$ are dense in the $[0,2\pi]$, so for $k>k_0$, there should be $y_{k,t+1}-y_{k,t}=\frac{2\pi}{k}+O(\frac{1}{n})$. How to show it in rigorous manner?