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From Tarski's plank problem to simultaneous approximation

Kupavskii, A. and Pach, János (2017) From Tarski's plank problem to simultaneous approximation. AMERICAN MATHEMATICAL MONTHLY, 124 (6). pp. 494-505. ISSN 0002-9890

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Abstract

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.

Item Type: Article
Subjects: Q Science / természettudomány > QA Mathematics / matematika
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 05 Feb 2018 13:01
Last Modified: 05 Feb 2018 13:01
URI: http://real.mtak.hu/id/eprint/73906

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