Measuring Anisotropy in Planar Sets

We define and discuss a pure mathematics formulation of an approach proposed in the physics literature to analysing anisotropy of fractal sets

The Hausdorff dimension of the visible sets of connected compact sets

For a compact subset K of the plane and a point x, we define the visible part of K from x to be the set K_x={u\in K : [x,u]\cap K={u}}. (Here [x,u] denotes the closed line segment joining x to u.) In this paper, we use energies to show that if K is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point x in the plane, the Hausdorff dimension of K_x is strictly less than the Hausdorff dimension of K. In fact, for almost every x, dim(K_x)\leq {1/2}+\sqrt{dim(K)-{3/4}}. We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than s+{1/2}+\sqrt{dim(K)-{3/4}}, for s>0.Comment: Approximately 40 pages with 6 figure

Points of middle density in the real line

A Lebesgue measurable set in the real line has Lebesgue density 0 or 1 at almost every point. Kolyada showed that there is a positive constant $\delta$ such that for non-trivial measurable sets there is at least one point with upper and lower densities lying in the interval $(\delta, 1-\delta)$. Both Kolyada and later Szenes gave bounds for the largest possible value of this $\delta$. In this note we reduce the best known upper bound, disproving a conjecture of Szenes

Visible parts and dimensions

We study the visible parts of subsets of n-dimensional Euclidean space: a point a of a compact set A is visible from an affine subspace K of Rn, if the line segment joining PK(a) to a only intersects A at a (here PK denotes projection onto K). The set of all such points visible from a given subspace K is called the visible part of A from K. We prove that if the Hausdorff dimension of a compact set is at most n−1, then the Hausdorff dimension of a visible part is almost surely equal to the Hausdorff dimension of the set. On the other hand, provided that the set has Hausdorff dimension larger than n − 1, we have the almost sure lower bound n − 1 for the Hausdorff dimensions of visible parts. We also investigate some examples of planar sets with Hausdorff dimension bigger than 1. In particular,we prove that for quasi-circles in the plane all visible parts have Hausdorff dimension equal to 1