Counting points in hypercubes and convolution measure algebras

D. Hajela, P. Seymour

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

It is shown that if A and B are non-empty subsets of {0, 1} n (for some nεN) then |A+B|≧(|A||B|)α where α=(1/2) log2 3 here and in what follows. In particular if |A|=2 n-1 then |A+A|≧3 n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 n-1 then |A+A|=3 n-1 if and only if the points of A lie on a hyperplane in n-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)α. The above results imply the following improvement of a result of Talagrand [7]: if X and Y are compact subsets of K (the Cantor set) with m(X), m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))α where m is the usual measure on K and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing that m is not concentrated on any proper Raikov system.

Original languageEnglish (US)
Pages (from-to)205-214
Number of pages10
JournalCombinatorica
Volume5
Issue number3
DOIs
StatePublished - Sep 1 1985
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

Keywords

  • AMS subject classification (1980): 10E05

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