From Tarski's Plank Problem to Simultaneous Approximation
From Tarski's Plank Problem to Simultaneous Approximation
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从塔斯基的普朗克问题到联立逼近
DOI：
10.4169/amer.math.monthly.124.6.494
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
J. Pach
中科院分区：
文献类型：
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作者：
A. Kupavskii;J. Pach
摘要 板（或板）是位于两个平行超平面之间的 d 维欧几里得空间的一部分。这些超平面之间的距离称为板的宽度。据推测，任何具有不同总宽度的无限板族的成员都可以平移，使得平移一起覆盖整个 d 维空间。我们证明了这个猜想的一个稍弱的版本，它可以被视为 Bang 定理的逆，也称为 Tarski 的木板问题。这个结果使我们能够解决 Makai 和 Pach 关于多项式联立逼近的古老猜想。如果平面上存在一系列点，其 x 坐标形成序列 S，则我们说正数的无限序列 S 控制最多 d 次多项式，使得每个最多 d 次多项式的图形至少在距其中一个点的垂直距离 1 内通过。我们证明序列 S 具有此性质当且仅当其元素的 d 次幂的倒数和发散。
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.