From Tarski's plank problem to simultaneous approximation

A slab (or plank) is the part of the d-dimensional Euclidean space that lies between two parallel hyperplanes. The distance between the these hyperplanes is called the width of the slab. It is conjectured that the members of any infinite family of slabs with divergent total width can be translated so that the translates together cover the whole d-dimensional space. We prove a slightly weaker version of this conjecture, which can be regarded as a converse of Bang's theorem, also known as Tarski's plank problem. This result enables us to settle an old conjecture of Makai and Pach on simultaneous approximation of polynomials. We say that an infinite sequence S of positive numbers controls all polynomials of degree at most d if there exists a sequence of points in the plane whose x-coordinates form the sequence S, such that the graph of every polynomial of degree at most d passes within vertical distance 1 from at least one of the points. We prove that a sequence S has this property if and only if the sum of the reciprocals of the dth powers of its elements is divergent. © The Mathematical Association of America

Controlling Lipschitz functions

Given any positive integers $m$ and $d$, we say the a sequence of points $(x_i)_{i\in I}$ in $\mathbb R^m$ is {\em Lipschitz-$d$-controlling} if one can select suitable values $y_i\; (i\in I)$ such that for every Lipschitz function $f:\mathbb R^m\rightarrow \mathbb R^d$ there exists $i$ with $|f(x_i)-y_i|<1$. We conjecture that for every $m\le d$, a sequence $(x_i)_{i\in I}\subset\mathbb R^m$ is $d$-controlling if and only if $$\sup_{n\in\mathbb N}\frac{|\{i\in I\, :\, |x_i|\le n\}|}{n^d}=\infty.$$ We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be $d$-controlling. We also prove the conjecture for $m=1$

Simultaneous Approximation of Polynomials

Let 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) <= x2 <=... is called P-d-controlling if there exist y(1), y(2),....is an element of R such that for every polynomial p is an element of P-d there exists an index i with |p(xi) - yi| <= 1. We settle a problem of Makai and Pach (1983) by showing that x(1) <= x(2) <= ... is P-d- controlling if and only if Sigma(infinity)(i=1) 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