Some remarks on a problem of Erdős and Graham
type: Item
instance of: scholarly article
title: Some remarks on a problem of Erdős and Graham
zbMATH Open document ID: 0682.10051
DOI: 10.1007/BF02170065
published in: Acta Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: For each strictly increasing sequence A of natural numbers let \[ P(A)=\{\sum^{\infty}_{k=0}\epsilon_ ka_ k :\quad \epsilon_ k=0\quad or\quad 1,\quad \sum^{\infty}_{k=0}\epsilon_ k<\infty \}. \] Let \(\alpha\), \(\beta\) be real numbers with \(\alpha\) /\(\beta\) irrational. Erdős and Graham conjectured that the sequence \(S=\{[\alpha],[\beta],...,[2^ n\alpha],[2^ n\beta],...\}\) contained all sufficiently large numbers or is said to be complete. This conjecture is reformulated and dyadic representation used to prove the partial form of the conjecture that if in the dyadic representation \(\alpha =\sum^{\infty}_{i=-k}\epsilon_ i2^{-i}\), where \(\epsilon_ i=1\) infinitely often, and if in the dyadic representation for \(\beta\), \(\epsilon_ i=1\) only finitely often, then S is complete. Other results which lend support to the conjecture are proved and stronger versions of the conjecture proposed \((\alpha /\beta \neq 2^{-n})\).
Mathematics Subject Classification ID: 11B99
Mathematics Subject Classification ID: 11D85
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 4116653
zbMATH Keywords: Erdős-Graham conjecture
zbMATH Keywords: complete sequence of integers
zbMATH Keywords: unique representability
zbMATH Keywords: dyadic representation
author: Norbert Hegyvári
reviewed by: M. M. Dodson
MaRDI profile type: Publication
cites work: On a conjecture of Erdös in additive number theory
description: scientific article; zbMATH DE number 4116653