Diophantine approximation with primes and powers of two
type: Item
instance of: scholarly article
title: Diophantine approximation with primes and powers of two
zbMATH Open document ID: 1038.11065
author: Scott T. Parsell
published in: The New York Journal of Mathematics
publication date: 2004-01-19T00:00:00Z
full work available at URL: https://eudml.org/doc/123546
full work available at URL: 9-19nf.htm
review text: Let \(\lambda_1\), \(\lambda_2\), \(\mu_1\), \(\mu_2\) be nonzero real numbers such that \(\lambda_1/\lambda_2\) is negative and irrational, and \(\lambda_1/\mu_1\) and \(\lambda_2/\mu_2\) are rational. Let \(\eta> 0\) be given. Then it is shown that there exists \(s_0\geq 2\) depending only on \(\lambda_1\), \(\lambda_2\), \(\mu_1\), \(\mu_2\) and \(\eta\) as follows. For any \(s> s_0\) and any real numbers \(\gamma\) and \(\mu_3,\mu_4,\dots, \mu_s\) the inequality \[ |\lambda_1 p_1+ \lambda_2 p_2+ \mu_1 2^{n_1}+\cdots+ \mu_2 2^{n_s}+ \gamma|< \eta \] has infinitely many solutions. This can be seen as a Diophantine approximation analogue of \textit{Yu. V. Linnik}'s result [Mat. Sb., N. Ser. 32(74), 3--60 (1953; Zbl 0051.03402)] on representations of integers using 2 primes and a bounded number of powers of 2. The proof combines the Davenport-Heilbronn circle method with a recent approach to Linnik's theorem, due to the reviewer and \textit{J.-C. Puchta} [Asian J. Math. 6, 535--566 (2002; Zbl 1097.11050)].
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 2028913
zbMATH Keywords: Goldbach-type theorems
zbMATH Keywords: Diophantine inequalities
zbMATH Keywords: applications of Hardy-Littlewood method
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2028913