Trigonometric series with gaps
type: Item
instance of: scholarly article
title: Trigonometric series with gaps
zbMATH Open document ID: 0544.10062
DOI: 10.1007/BF01903989
author: I. M. Mikheev
published in: Analysis Mathematica
publication date: 1983-01-01T00:00:00Z
review text: Let a, \(m_ 1,...,m_ n\in {\mathbb{Z}}\) and \(m_ 1>0\), \(m_ k>m_ 1+...+m_{k-1}\), \(k=1,...,n\). Then the set \(\{a+\epsilon_ 1m_ 1+...+\epsilon_ nm_ n,\quad \epsilon_ i=0\quad or\quad 1,\quad i=1,...,n\}\) is called a tracing segment of length \(2^ n\). For instance, any arithmetic progression of \(2^ n\) terms is a tracing segment of length \(2^ n\). The author investigates the sequences containing arbitrarily long tracing segments, and obtains some general results. As a consequence he proves that the sequence of primes contains arbitrarily long tracing segments. These results are then used in order to give an affirmative answer to a conjecture of \textit{W. Rudin} [J. Math. Mech. 9, 203-227 (1960; Zbl 0091.058)] concerning the density of the so- called \(\Lambda\) (p) sequences, in the case \(0<p\leq 2\).
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 42A32
zbMATH DE Number: 3865420
zbMATH Keywords: regular sets
zbMATH Keywords: trigonometric series with gaps
zbMATH Keywords: sequences containing arbitrarily long tracing segments
reviewed by: A. Perelli
MaRDI profile type: Publication
cites work: On Sets of Integers Which Contain No Three Terms in Arithmetical Progression
cites work: On Some Sequences of Integers
cites work: On a Problem of Sidon in Additive Number Theory, and on some Related Problems
cites work: On Non-Averaging Sets of Integers
cites work: On Certain Sets of Integers
cites work: Q3265794
cites work: On sets of integers containing no four elements in arithmetic progression
cites work: On sets of integers containing k elements in arithmetic progression
full work available at URL: https://doi.org/10.1007/bf01903989
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description: scientific article; zbMATH DE number 3865420