Restricted families of projections and random subspaces. (English) Zbl 1401.28012

A fundamental problem in fractal geometry is to determine how the projections affect dimension. In the last decay, there has been great interest in understanding the fractal dimension of projections of sets and measures. The first significant work in this area was the result of J. M. Marstrand [Proc. Camb. Philos. Soc. 50, 198–202 (1954; Zbl 0055.05102)] who showed that for any Borel set, \(E\) of the plane, almost every orthogonal projection \(\pi_V\) onto a line \(V\), \[ \dim_H(\pi_V E)=\min\big(\dim_H E,1\big), \] where \(\dim_H\) denotes the usual Hausdorff dimension. This result was later generalized to higher dimensions by R. Kaufman [Mathematika 15, 153–155 (1968; Zbl 0165.37404)] and P. Mattila [Ann. Acad. Sci. Fenn., Ser. A I, Math. 1, 227–244 (1975; Zbl 0348.28019)] and they obtain similar results for the Hausdorff dimension of a measure. In the paper under review, the author studied the restricted families of orthogonal projections in \(\mathbb{R}^3\) and showed that there are families of random subspaces which admit Marstrand-Mattila type projection result (Theorem 3).


28A80 Fractals
28A78 Hausdorff and packing measures
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