Let $f:\mathbb{T}^m \to \mathbb{R}$ is a function of bounded variation(BV). Let $D=\{\boldsymbol{p}_i,i=1,2,3\ldots\}$ be a countable dense subset of $(0,1)^m$. Let $E_n, n = 1,2,3\ldots$ be a sequence of sets defined as $E_n = \{\boldsymbol{p_i}/\boldsymbol{p_i}\in D, i = 1,2,3\ldots n\}$.

Define the mesh norm of the data points set $E_n$ over the domain $\Omega = (0,1)^m$ as \begin{equation}\label{mesh_norm} \zeta_n = \sup\limits_{\boldsymbol{x}\in\Omega}\inf\limits_{\boldsymbol{p}\in E_n}\|\boldsymbol{x}-\boldsymbol{p}\|_2 \end{equation}

As $D$ is dense we know that $$\lim\limits_{n\to\infty}\zeta_n = 0$$

Also as $f$ is BV, we have $$ \lim\limits_{n\to\infty}\left(\|f\|^2_{L^2(\mathbb{T}^m)}-\frac{1}{n}\sum\limits_{i=1}^n\left(f(\boldsymbol{p_i})\right)^2\right) = 0$$

Question

I am looking for an estimate between the above two expressions for sufficiently large $n$. That is how does LHS of the above expression decay as $\zeta_n$ decays with $n$.

Something like, for sufficiently large $n$, $$\left(\|f\|^2_{L^2(\mathbb{T}^m)}-\frac{1}{n}\sum\limits_{i=1}^n\left(f(\boldsymbol{p_i})\right)^2\right) \le h(\zeta_n)$$

I want to find such a best possible $h$.

PS: Note $h$ should be such that $\lim\limits_{n\to\infty}h(\zeta_n) = 0$