From Tarski's plank problem to simultaneous approximation

TL;DR

This paper extends Tarski's plank problem by establishing conditions under which divergent slabs cover space and applies these results to characterize sequences that control all polynomials of a given degree, settling a longstanding conjecture.

Contribution

It proves a new condition for the coverage of space by divergent slabs and characterizes sequences controlling polynomial approximation, resolving an old conjecture.

Findings

Coverage of space by slabs with certain width conditions.

Characterization of sequences controlling polynomial approximation.

Resolution of Makai and Pach's conjecture.

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.