Hausdorff Dimension Exceptional Set Estimates for Projections, Sections and Intersections

Abstract

This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family P λ : R n → R m , λ ∈ Λ, of orthogonal projections which guarantee that the Hausdorff dimension formula dim A ∩ P −1 λ {u} = s − m holds generically for measurable sets A ⊂ R n with positive and finite s-dimensional Hausdorff measure, s > m. As an application we prove for measurable sets A, B ⊂ R n with positive s-and t-dimensional measures that if s + (n − 1)t/n > n, then dim A ∩ (g(B) + z) ≥ s + t − n for almost all rotations g and for positively many z ∈ R n . We shall also give an application on the estimates of the dimension of the set of exceptional rotations.