2015/11/25 by Andrey B. Kupavskii, János Pach, Kupavskii, Andrey B. +1 · 1 citation
Computer Science · Mathematics · #52C17 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG) #cs.DM #math.CO #math.MG #msc:52C17
paper · pdf · doi:10.48550/arxiv.1511.08111
arxiv created 2016/07/20 · arxiv updated 2017/12/01
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 S1, S2,… whose total width is divergent have suitable translates that altogether cover ℝd. We show that this statement is true if the widths of the slabs, w1, w2,…, satisfy the slightly stronger condition \limsupn→∞(w1+w2+…+wn)/(log(1/wn))>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 x1≤ x2≤… \em controls all polynomials of degree at most d if there exist y1, y2,…∈ℝ such that for every polynomial p of degree at most d, there exists an index i with |p(xi)-yi|≤ 1. We prove that a sequence has this property if and only if ∑i=1∞(1)/(xid) is divergent. This settles an old conjecture of Makai and Pach.