Let f : R² → R be a function depending on two variables, and consider the following two claims: (1) At most points the value of the function f can be well approximated by averaging f over small disks containing the point, and (2) At most points, the value of f can be well approximated by averaging the function over any short enough line segments passing through the point. It turns out that 1 is true; this result is classical. We discuss it in Chapter 1. Discussion of 2 requires that we say something about the directions in which these short line segments are allowed to point. Once we specify a certain set of directions, we can ask whether 2 is true when the line segments are only allowed to point in one of these specified directions. This dissertation categorizes sets of directions for which statement 2 is false. The author uses probabilistic methods to show that certain collections of rectangles have unusual overlapping properties. It turns out that such collections of rectangles can be constructed whenever the set of directions in question has a tree-like structure.

Specifically, in Chapter 2 we give a classification of sets of directions Ω such that the associated directional maximal operator M _Ω is bounded on L^p. We show that M_Ω is bounded if and only if Ω is essentially a lacunary set.

In Chapter 3 we present a result about the boundedness of a simple maximal operator on L^p when the choice of averaging direction depends only on one variable.

Finally, in Chapter 4 we give an application of some ideas used in the first section to a problem in geometric measure theory. We give a lower bound on the Favard length of the nth generation 4-corner planar Cantor set. Our estimate of [special characters omitted] beats the value of [special characters omitted] that was expected based on consideration of random Cantor sets.