Counting points in hypercubes and convolution measure algebras

Abstract

It is shown that ifA andB are non-empty subsets of {0, 1}n (for somenεN) then |A+B|≧(|A||B|)α where α=(1/2) log2 3 here and in what follows. In particular if |A|=2n-1 then |A+A|≧3n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2n-1 then |A+A|=3n-1 if and only if the points ofA lie on a hyperplane inn-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]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.

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Hajela, D., Seymour, P. Counting points in hypercubes and convolution measure algebras. Combinatorica 5, 205–214 (1985). https://doi.org/10.1007/BF02579363

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AMS subject classification (1980)

  • 10 E 05