2023/10/11 by Mattila, Pertti
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2310.07538
This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family Pλ:ℝn→ℝm, λ∈ Λ, of orthogonal projections which guarantee that the Hausdorff dimension formula dim A∩ Pλ-1\u\=s-m holds generically for measurable sets A⊂ℝn with positive and finite s-dimensional Hausdorff measure, s>m. As an application we prove for measurable sets A,B⊂ℝn 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∈ℝn. We shall also give an application on the estimates of the dimension of the set of exceptional rotations.