Counting points in hypercubes and convolution measure algebras
type: Item
instance of: scholarly article
title: Counting points in hypercubes and convolution measure algebras
zbMATH Open document ID: 0597.05003
DOI: 10.1007/BF02579363
author: Paul Seymour
author: D. J. Hajela
published in: Combinatorica
publication date: 1985-01-01T00:00:00Z
review text: Let A and B be non-empty subsets of \(\{0,1\}^ n\), \(\alpha =(1/2)\log_ 2 3\). Then it is shown that the sumset \(A+B\) has cardinality \(| A+B| \geq (| A| | B|)^{\alpha},\) as had been conjectured by Erdős in the case \(A=B\). Necessary and sufficient conditions for equality are also given. The following result is proved as an application: Let \(\lambda\) be Lebesgue measure, and \(\mu\) Haar measure on the Cantor group \(D=\{0,1\}^{{\mathbb{N}}}\), identified with the Cantor set in [0,1]. If \(X,Y\subseteq D\) are analytic sets with \(\mu(X)\mu(Y)>0\) then \(\lambda(X+y)\geq 2\mu(X)^{\alpha}\mu (Y)^{\alpha}.\)
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11P99
zbMATH DE Number: 3961615
zbMATH Keywords: extremal problem
zbMATH Keywords: Raikov systems
zbMATH Keywords: sumset
zbMATH Keywords: cardinality
zbMATH Keywords: Lebesgue measure
zbMATH Keywords: Haar measure
zbMATH Keywords: Cantor set
reviewed by: Bodo Volkmann
MaRDI profile type: Publication
cites work: Raikov Systems and Radicals in Convolution Measure Algebras
cites work: Q3881310
cites work: A Problem in Combinatorial Geometry
cites work: An inequality conjectured by Hajela and Seymour arising in combinatorial geometry
description: scientific article; zbMATH DE number 3961615