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.
Similar content being viewed by others
References
G. Brown andW. Moran,L. M. S. Research Symposium on Functional Analysis and Stochastic Processes, Durham (England), August 1974.
G. Brown andW. Moran, Raikov Systems an Radicals in Convolution Measure Algebras,J. London Math. Soc.,28 (1983), 531–542.
C. Graham andO. C. McGehee,Essays in Commutative Harmonic Analysis, Springer-Verlag, 1979.
R. Hall, A Problem in Combinatorial Geometry,J. London Math. Soc., (2),12 (1976), 315–319.
H. Landau, B. Logan andL. Shepp, An Inequality Conjectured by Hajela and Seymour Arising in Combinatorial Geometry,Combinatorica,5 (1985), 337–342.
G. Pólya andG. Szegő.Problems and Theorems in Analysis, Springer-Verlag, 1976.
M. Talagrand. Solution d’un Probleme de R. Haydon,Publications du Department de Mathematiques Lyon, 12–2 (1975), 43–46.
D. Woodall, A Theorem on Cubes.Mathematika24 (1977), 60–62.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hajela, D., Seymour, P. Counting points in hypercubes and convolution measure algebras. Combinatorica 5, 205–214 (1985). https://doi.org/10.1007/BF02579363
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02579363