Analysis of Minkowski contents of fractal sets and applications.

Abstract

Using fractal sets and Minkowski contents we extend the repertoire of Lebesgue integrable functions to those with large singular sets. A new method of constructing fractal sets is proposed, using a class of absolutely continuous functions, called swarming functions. We obtain bounds on Minkowski contents of fractals in terms of two natural parameters contained in $[-\infty,\infty]$, called the upper and lower dispersions of the fractal. Assuming that upper and lower box dimensions of a fractal are equal, we show that if the difference of dispersions is sufficiently large, then the set is not Minkowski measurable. Fractals with nondegenerate $d$-dimensional Minkowski contents (i.e., contained in $(0,\infty)$) are characterized as those with nondegenerate dispersions (i.e., contained in $(-\infty,\infty)$). The Weierstrass function, a class of affine fractal functions and a class of McMullen's sets have nondegenerate Minkowski contents. Also some classes of spirals of focus and limit cycle type in the plane are shown to be Minkowski measurable. Using swarming functions we can easily construct fractal sets with maximally separated lower and upper box dimensions, and a pair of fractal sets with maximal instability of lower box dimension with respect to union. We also study gauge functions associated with fractals having degenerate Minkowski contents, and obtain new integrability criteria for a class of singular integrals.