Counting points in hypercubes and convolution measure algebras


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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  1. [1]

    G. Brown andW. Moran,L. M. S. Research Symposium on Functional Analysis and Stochastic Processes, Durham (England), August 1974.

  2. [2]

    G. Brown andW. Moran, Raikov Systems an Radicals in Convolution Measure Algebras,J. London Math. Soc.,28 (1983), 531–542.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    C. Graham andO. C. McGehee,Essays in Commutative Harmonic Analysis, Springer-Verlag, 1979.

  4. [4]

    R. Hall, A Problem in Combinatorial Geometry,J. London Math. Soc., (2),12 (1976), 315–319.

    MATH  Article  Google Scholar 

  5. [5]

    H. Landau, B. Logan andL. Shepp, An Inequality Conjectured by Hajela and Seymour Arising in Combinatorial Geometry,Combinatorica,5 (1985), 337–342.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    G. Pólya andG. Szegő.Problems and Theorems in Analysis, Springer-Verlag, 1976.

  7. [7]

    M. Talagrand. Solution d’un Probleme de R. Haydon,Publications du Department de Mathematiques Lyon, 12–2 (1975), 43–46.

    MathSciNet  Google Scholar 

  8. [8]

    D. Woodall, A Theorem on Cubes.Mathematika24 (1977), 60–62.

    MathSciNet  Article  Google Scholar 

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Hajela, D., Seymour, P. Counting points in hypercubes and convolution measure algebras. Combinatorica 5, 205–214 (1985).

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