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 Pλ:Rn→Rm,λ∈Λ$P_{\lambda }:\mathbb {R}^n\rightarrow \mathbb {R}^m, \lambda \in \Lambda$, of orthogonal projections which guarantee that the Hausdorff dimension formula dimA∩Pλ−1{u}=s−m$\dim A\cap P_{\lambda }^{-1}\lbrace u\rbrace =s-m$ holds generically for measurable sets A⊂Rn$A\subset \mathbb {R}^{n}$ with positive and finite s$s$‐dimensional Hausdorff measure, s>m$s>m$. As an application we prove for Borel sets A,B⊂Rn$A,B\subset \mathbb {R}^{n}$ with positive s$s$‐ and t-dimensional$t{\text{-dimensional}}$ measures that if s+(n−1)t/n>n$s + (n-1)t/n > n$, then dimA∩(g(B)+z)⩾s+t−n$\dim A\cap (g(B)+z) \geqslant s+t - n$ for almost all rotations g$g$ and for positively many z∈Rn$z\in \mathbb {R}^{n}$. We shall also give an application to the estimates of the dimension of the set of exceptional rotations. [ABSTRACT FROM AUTHOR]