From asymptotic distribution and vague convergence to uniform convergence, with numerical applications

Let $\{{\mathrm{\Lambda }}_n=\{{\lambda }_{1,n},\dots ,{\lambda }_{d_n,n}\}{\}}_n$ be a sequence of finite multisets of real numbers such that $d_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$, and let $f:\mathrm{\Omega }\subset {\mathbb{R}}^d\to \mathbb{R}$ be a Lebesgue measurable function defined on a domain $\mathrm{\Omega }$ with $0<{\mu }_d(\mathrm{\Omega })<\mathrm{\infty }$, where ${\mu }_d$ is the Lebesgue measure in ${\mathbb{R}}^d$. We say that $\{{\mathrm{\Lambda }}_n{\}}_n$ has an asymptotic distribution described by $f$, and we write $\{{\mathrm{\Lambda }}_n{\}}_n\sim f$, if $$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{d_n}\sum _{i=1}^{d_n}F({\lambda }_{i,n})=\frac{1}{{\mu }_d(\mathrm{\Omega })}{\int }_{\mathrm{\Omega }}F(f(\mathit{x}))\mathrm{d}\mathit{x}(\ast )$$ for every continuous function $F$ with bounded support. If ${\mathrm{\Lambda }}_n$ is the spectrum of a matrix $A_n$, we say that $\{A_n{\}}_n$ has an asymptotic spectral distribution described by $f$ and we write $\{A_n{\}}_n{\sim }_{\lambda }f$. In the case where $d=1$, $\mathrm{\Omega }$~is a bounded interval, ${\mathrm{\Lambda }}_n\subseteq f(\mathrm{\Omega })$ for all $n$, and $f$ satisfies suitable conditions, Bogoya, Böttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to $0$ of the difference between the properly sorted vector $[{\lambda }_{1,n},\dots ,{\lambda }_{d_n,n}]$ and the vector of samples $[f(x_{1,n}),\dots ,f(x_{d_n,n})]$, i.e., $$\underset{n\to \mathrm{\infty }}{\lim }\underset{i=1,\dots ,d_n}{\max }\mathrm{\mid }f(x_{i,n})-{\lambda }_{{\tau }_n(i),n}\mathrm{\mid }=0,(\ast \ast )$$ where $x_{1,n},\dots ,x_{d_n,n}$ is a uniform grid in $\mathrm{\Omega }$ and ${\tau }_n$ is the sorting permutation. We extend this result to the case where $d\ge 1$ and $\mathrm{\Omega }$ is a Peano–Jordan measurable set (i.e., a bounded set with ${\mu }_d(\mathrm{\partial }\mathrm{\Omega })=0$). See the rest of the abstract in the manuscript.