Uniformly distributed measures in Euclidean spaces

Abstract

of Marstrand [6] according to which the existence of non-trivial s-dimensional densities (of a measure in R n ) implies that s is an integer. Our results, however, do not seem to be strong enough to show the main result of [9] that measures having s-dimensional density are s rectifiable, because they do not give any information about the behaviour of X at infinity. In addition to the analyticity result mentioned above, we also show that X is an algebraic variety provided that it is bounded and obtain more precise results in the special cases n = 1 and n = 2. For various reasons, including the political changes in Europe, this note has not been published for many years, although our main approach has become known and used in the literature. We would like to thank the Max Planck Institute for bringing us together and thus