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Suppose that $\Sigma \subset \mathbb{C}$ is compact and symmetric about thereal axis and is a finite union of rectangles and real intervals withtransfinite diameter $d_{\Sigma}>1$. Suppose that $\mu$ is a H\"olderarithmetic probability distribution on $\Sigma$ defined in~\cite{Naser} withlogarithmic energy \[ I(\mu):=\int\int \log|z_1-z_2|d\mu(z_1)d\mu(z_2). \] Let$\mathcal{Z}=\{z_1,\dots,z_n\}$ be a generic sampling of $n$ points accordingto $\mu$. Let $m<n$. We prove that there are$e^{\frac{n^2}{2}I(\mu)+O(mn+m^2+n^{2-\delta})}$ irreducible polynomial ofdegree $m+n$ with roots $\alpha_1,\dots,\alpha_{m+n}$ such that$|z_i-\alpha_i|<d_{\Sigma}^{-m(1-O(n^{-\eta}))}$ for $1\leq i\leq n$ and$\alpha_i \in \Sigma$ for every $1\leq i\leq m+n$, where $\delta>0$ and$\eta>0$ depend only on $\mu$. Given any finite field $\mathbb{F}_q$ and anyinteger $n$, we give an asymptotic formula for the number of isogeny classes ofordinary Abelian verities of dimension $n$ with prescribed Frobenius eigenvalueangles in small intervals.

Conjugate algebraic integers and 2D-Coulomb's law