Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes

Abstract

We present strong versions of Marstrand's projection theorems and other related theorems. For example, if $E$ is a plane set of positive and finite $s$-dimensional Hausdorff measure, there is a set $X$ of directions of Lebesgue measure $0$, such that the projection onto any line with direction outside $X$, of any subset $F$ of $E$ of positive $s$-dimensional measure, has Hausdorff dimension min{1, $s$}, i.e. the set of exceptional directions is independent of $F$. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.