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A concavity property for the measure of product sets in groups

Imre Ruzsa (1992)

Fundamenta Mathematicae

Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality μ ̲ ( A B ) ≥ m i n ( μ ̲ ( A ) + μ ̲ ( B ) , μ ( G ) ) for unimodular G.

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube I M = [ 0 , 1 ) M , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on I M ; (3) the measures have the convolution property that μ ∗ L p ⊆ L p + ε for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then μ ∗ L p ⊆ L q for any measure μ in our class.

A note on intersections of non-Haar null sets

Eva Matoušková, Miroslav Zelený (2003)

Colloquium Mathematicae

We show that in every Polish, abelian, non-locally compact group G there exist non-Haar null sets A and B such that the set {g ∈ G; (g+A) ∩ B is non-Haar null} is empty. This answers a question posed by Christensen.

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