On the structure of a random sum-free set
type: Item
instance of: scholarly article
title: On the structure of a random sum-free set
zbMATH Open document ID: 0617.60001
DOI: 10.1007/BF00960072
author: Peter J. Cameron
publication date: 1987-01-01T00:00:00Z
review text: There is a natural probability measure on the set \(\Sigma\) of all sum- free sets of natural numbers. If we represent such a set by its characteristic function s, then the zero-one random variables s(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of \(\Sigma\) into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218..., and its members almost surely have density 1/4.
Mathematics Subject Classification ID: 60A10
Mathematics Subject Classification ID: 11K38
zbMATH DE Number: 4001068
zbMATH Keywords: sum-free sets of natural numbers
zbMATH Keywords: decomposition
MaRDI profile type: Publication
cites work: Cyclic automorphisms of a countable graph and random sum-free sets
cites work: Correlation inequalities on some partially ordered sets
cites work: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions
cites work: Q3956129
cites work: Q3333071
cites work: On sets of integers containing k elements in arithmetic progression
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4001068