2019/01/30 by K. J. Falconer, Falconer, Kenneth J. · 4 citations
Mathematics · #28A80 #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1901.11014
openalex publication_date 2019/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Dimension profiles were introduced in [8,11] to give a formula for the\nbox-counting and packing dimensions of the orthogonal projections of a set\nRn onto almost all m-dimensional subspaces. However, these definitions of\ndimension profiles are indirect and are hard to work with. Here we firstly give\nalternative definitions of dimension profiles in terms of capacities of E\nwith respect to certain kernels, which lead to the box-counting and packing\ndimensions of projections fairly easily, including estimates on the size of the\nexceptional sets of subspaces where the dimension of projection is smaller the\ntypical value. Secondly, we argue that with this approach projection results\nfor different types of dimension may be thought of in a unified way. Thirdly,\nwe use a Fourier transform method to obtain further inequalities on the size of\nthe exceptional subspaces.\n