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Coin tossing and sum sets

Part of: Abstract harmonic analysis Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  09 April 2009

Gavin Brown  and
John H. Williamson
Gavin Brown
Affiliation:
Department of Pure MathematicsUniversity of New South WalesKensington, 2033, N.S.W., Australia
John H. Williamson
Affiliation:
Department of Pure MathematicsUniversity of New South WalesKensington, 2033, N.S.W., Australia
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Abstract

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We consider the distribution μ of numbers whose binary digits are generated from infinitely many tosses of a biased coin. It is shown that, if E has positive μ measure, then some n-fold sum of E with itself must contain an interval. This contrasts with the known result that all convolution powers of μ are singular.

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 43A05: Measures on groups and semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 2 , October 1987 , pp. 211 - 219
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Brown, G., Keane, M., Moran, W., and Pearce, C., ‘A combinatorial inequality with applications to Cantor measures,’ preprint.Google Scholar
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[3]Brown, G., Moran, W., and Tijdeman, R., ‘Riesz products are basic measures’, J. London Math. Soc. (2) 30 (1984), 105–109.CrossRefGoogle Scholar
[4]Hewitt, E. and Stromberg, K., Real and Abstract Analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1965).Google Scholar
[5]Oberlin, D. M., ‘The size of sum sets, II,’ preprint.Google Scholar