Box and packing dimensions of projections and dimension profiles (Q2709860)

Let \(0\leq m\leq n\), \(E\) be an analytic subset of \( {\mathbb R}^n\), \(P_VE\) denote the orthogonal projection of \(E\) onto an \(m\)-dimensional subspace \(V\). \textit{K. J. Falconer} [Mathematika 29, 109--115 (1982; Zbl 0477.28004)], \textit{J. M. Marstrand} [Proc. Lond. Math. Soc., III. Ser. 4, 257--302 (1954; Zbl 0056.05504)] and \textit{P. Mattila} [Ann. Acad. Sci. Fenn., Ser. A I 1, 227--244 (1975; Zbl 0348.28019)] studied the Hausdorff dimension of the orthogonal projections of \(E\). \textit{M. Järvenpää} [Ann. Acad. Sci. Fenn., Ser. A I, Dissertat. 99 (1994; Zbl 0811.28003)] studied the packing and upper box dimensions of the orthogonal projections of \(E\) and Järvenpää's results were improved and developed by \textit{K. J. Falconer} and \textit{J. D. Howroyd} [Math. Proc. Camb. Philos. Soc. 119, No. 2, 287-295 (1996; Zbl 0846.28004)] and \textit{K. J. Falconer} and \textit{P. Mattila} [Math. Proc. Camb. Philos. Soc. 119, No. 4, 695--713 (1996; Zbl 0867.28005)]. In 1997, \textit{K. J. Falconer} and \textit{J. D. Howroyd} [Math. Proc. Camb. Philos. Soc. 121, No. 2, 269--286 (1997; Zbl 0881.28002)] further studied the packing dimension of the projections of \(E\) and packing dimension profiles by defining a packing-type dimension in terms of a `potential' obtained by convolving measures with certain kernels and obtained more results. NEWLINENEWLINEIn this paper, the author presents the direct analogues for upper and lower box dimensions of the results that have been proved by Falconer and Howroyd for packing dimension. The author defines upper and lower box dimension profiles that are closely related to the box dimensions of the orthogonal projections of \(E\) and obtain that the orthogonal projection of \(E\) onto almost all \(m\)-dimensional subspaces has upper box dimension \(B\text{-}\overline{\dim}_m E\) and lower box dimension \(B\text{-}\underline{\dim}_m E\). In addition, the connection between the upper box dimension profile and packing dimension profile is discussed.