On Kahane's ultraflat polynomials
type: Item
instance of: scholarly article
title: On Kahane's ultraflat polynomials
zbMATH Open document ID: 1170.42001
DOI: 10.4171/JEMS/163
author: Jean Bourgain
author: Enrico Bombieri
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2009-06-16T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: The paper at hand is a serious piece of work. The topic is, unsurprisingly, Kahane's ultraflat polynomials and the authors address two main issues: improving the error term and explicit construction. The authors' main result is that for every \(\epsilon>0\) and positive integer \(N\) there is a function \(f:\{0,\dots,N\}\rightarrow S^1\) such that \[ |\widehat{f}(\theta)| = \sqrt{n} + O_\epsilon(n^{1/2-1/9 + \epsilon}) \text{ for all } \theta \in \mathbb T. \] Moreover, this functions is effectively constructable meaning that all the coefficients are given explicitly in terms of elementary fnctions and Legendre or Jacobi symbols evaluated at square-free numbers in prescribed intervals. This error term considerably improves the previous best and the problem of explicit constructability was previously unresolved. In fact the \(O_\epsilon(n^\epsilon)\) can be improved to a power of \(\log n\) if one isn't concerned about constructability and \(\exp(O(\log n / \log \log n))\) otherwise. There is an extensive introduction and plan of the paper which makes clear how the results proceed and which ingredients are involved; this is a very welcome and arguably necessary part of the paper.
reviewed by: Tom Sanders
Mathematics Subject Classification ID: 42A05
zbMATH DE Number: 5565394
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: probabilistic methods
zbMATH Keywords: exponential sums
MaRDI profile type: Publication
OpenAlex ID: W2064819128
Recommended article: On Bernstein's inequality and Kahane's ultraflat polynomials
Recommended article: Polynomials on Köthe echelon spaces
Recommended article: \(K\)-theoretic polynomials
Recommended article: On the Krall-type polynomials
Recommended article: Q4501438
Recommended article: On a conjecture concerning Kloosterman polynomials
Recommended article: On a certain value of the Kauffman polynomial
Recommended article: Kac Polynomials for Canonical Algebras
Recommended article: Q4015656
Recommended article: On Kronecker polynomials
description: scientific article; zbMATH DE number 5565394