Hausdorff dimension of plane sections and general intersections

This paper extends some results of Mattila ( J. Fractal Geom . 66 (2021) 389–401 and Ann. Acad. Sci. Fenn. A Math . 42 (2017) 611–620), in particular, removing assumptions of positive lower density. We give conditions on a general family , of orthogonal projections which guarantee that the Hausdorff dimension formula holds generically for measurable sets with positive and finite ‐dimensional Hausdorff measure, . As an application we prove for Borel sets with positive ‐ and measures that if , then for almost all rotations and for positively many . We shall also give an application to the estimates of the dimension of the set of exceptional rotations.