From Tarski's plank problem to simultaneous approximation

Abstract

A {\em slab} (or plank) of width $w$ is a part of the $d$-dimensional space that lies between two parallel hyperplanes at distance $w$ from each other. It is conjectured that any slabs $S_1, S_2,\ldots$ whose total width is divergent have suitable translates that altogether cover $\mathbb{R}^d$. We show that this statement is true if the widths of the slabs, $w_1, w_2,\ldots$, satisfy the slightly stronger condition $\limsup_{n\rightarrow\infty}\frac{w_1+w_2+\ldots+w_n}{\log(1/w_n)}>0$. This can be regarded as a converse of Bang's theorem, better known as Tarski's plank problem. We apply our results to a problem on simultaneous approximation of polynomials. Given a positive integer $d$, we say that a sequence of positive numbers $x_1\le x_2\le\ldots$ {\em controls} all polynomials of degree at most $d$ if there exist $y_1, y_2,\ldots\in\mathbb{R}$ such that for every polynomial $p$ of degree at most $d$, there exists an index $i$ with $|p(x_i)-y_i|\leq 1.$ We prove that a sequence has this property if and only if $\sum_{i=1}^{\infty}\frac{1}{x_i^d}$ is divergent. This settles an old conjecture of Makai and Pach.