Simultaneous Approximation of Polynomials

Abstract

Let $\mathcal{P}_d$ denote the family of all polynomials of degree at most d in one variable x, with real coefficients. A sequence of positive numbers $x_1\le x_2\le \ldots$ is called $\mathcal{P}_d$-controlling if there exist $y_1, y_2,\ldots \in \mathbb {R}$ such that for every polynomial $p\in \mathcal{P}_d$ there exists an index i with $|p(x_i)-y_i|\le 1.$ We settle a problem of Makai and Pach (1983) by showing that $x_1\le x_2\le \ldots$ is $\mathcal{P}_d$-controlling if and only if $\sum _{i=1}^{\infty }\frac{1}{x_i^d}$ is divergent. The proof is based on a statement about covering the Euclidean space with translates of slabs, which is related to Tarski’s plank problem.