Arithmetic progressions in lacunary sets
type: Item
instance of: scholarly article
title: Arithmetic progressions in lacunary sets
zbMATH Open document ID: 0632.10052
DOI: 10.1216/RMJ-1987-17-3-587
published in: Rocky Mountain Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: One of the well-known Erdős conjectures says: If for a set A of positive integers the series \(\sum_{a\in A}a^{-1}\) diverges then A contains a k-term arithmetic progression for all \(k\geq 1\). In the paper some observations regarding this conjecture are made and several special cases of the conjecture are proved. For instance, the following statement is equivalent to the above conjecture: For each positive integer k there exists T such that if \(\sum_{a\in A}a^{-1}>T\) then A contains a k-term arithmetic progression. Or, if A is M-lacunary (i.e. if the sequence of differences of consecutive terms of A is non-decreasing and tends to infinity) and \(\sum_{a\in A}a^{-1}=\infty,\) then A satisfies the Erdős conjecture, etc.
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 4027600
zbMATH Keywords: Szemeredi's theorem
zbMATH Keywords: Erdős conjecture
zbMATH Keywords: lacunary sequence
zbMATH Keywords: k-term arithmetic progression
author: Tom Brown
author: Allen R. Freedman
reviewed by: Štefan Porubský
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4027600