Thin Sets with Fat Shadows: Projections of Cantor Sets

Abstract

A Cantor set is a nonempty, compact, totally disconnected, perfect subset of IR. Now, the set being totally disconnected means that it is scattered about like a “dust”. If you shine light on a clump of dust floating in the air, the shadow of this dust will look like a bunch of spots on the wall. You would be very surprised if you saw that the shadow was a filled-in shape (like a rabbit, say!). That would be pretty unbelievable. So, is this possible? We can think of the projection of a Cantor set onto a subspace as the shadow on that subspace. Is it possible that a cloud of dust (a Cantor set) could have a shadow (projection) which is “filled-in” (homeomorphic to the n − 1 dimensional unit ball)? The answer is YES! In fact, it is possible to have the shadow in every direction be “filled-in”! In this note we give an example of a simple construction of a Cantor subset of the unit square whose projection in every direction is a line segment. This construction can easily be generalized to n dimensions.