Nonstandard approximations to Besicovitch sets

Abstract

A nonstandard approximation to a Besicovitch set in R2 is constructed. It is shown that this approximation easily yields the correct upper Minkowski dimension in the two-dimensional case through a direct computation. By constructing an approximation to a Besicovitch set in three dimensions, we use a theorem of Guth and Katz to obtain an upper bound on the number of intersections of line segments in nonstandard terms.