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A note on the density of k-free polynomial sets, Haar measure and global fields
Quaestiones Mathematicae ( IF 0.6 ) Pub Date : 2021-07-16 , DOI: 10.2989/16073606.2021.1945701
Luca Demangos 1 , Ignazio Longhi 2
Affiliation  

Abstract

In this work we investigate the general relation between the density of a subset of the ring of integers D of a general global field and the Haar measure of its closure in the profinite completion . We then study a specific family of sets, the preimages of k-free elements (for any given k ∈ ℕ\{0, 1}) via one variable polynomial maps, showing that under some hypotheses their asymptotic density always exists and it is precisely the Haar measure of the closure in of their set.



中文翻译:

关于 k-free 多项式集、Haar 测度和全局场的密度的注释

摘要

在这项工作中,我们研究了一般全局场的整数环D的子集的密度与其在有限补全中的封闭性的 Haar 度量之间的一般关系。然后,我们通过一个变量多项式映射研究了一个特定的集合族,k自由元素的原像(对于任何给定的k ∈ ℕ\{0, 1}),表明在某些假设下,它们的渐近密度总是存在的,并且正是他们的集合中封闭的 Haar 度量。

更新日期:2021-07-16
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