Abstract
One of the main concepts of geometric measure theory is that of a rectifiable set, and one of its main objectives is to characterize them by means of geometric or analytic properties. One possible characterization is in terms of densities. Given integers 0 < d < n and a set E of Rn, the lower d-density of E at a point x of Rn is defined as the lower limit as r tends to 0 of the quotient between the d-dimensional Hausdorff measure of E ∩ B(x,r), which we denote H(E ∩ B(x,r)), and (2r)^d. In 1938, Besicovitch proved that given a set E of the plane R2, if the lower 1-density of E is strictly greater than 3/4 at almost all of its points, then E is 1-rectifiable. He also gave an example of a purely 1-unrectifiable set of the plane R2, with lower density equal to 1/2 at almost all of its points.
In this presentation, we present a generalization to arbitrary dimensions of the example originally introduced by Besicovitch. In this way, we obtain a purely d-unrectifiable set in Rd+1 with lower d-density equal to 1/2 at almost all of its points. This establishes the lower bound 1/2 for the minimum value S such that if the lower d-density of a set E is strictly greater than S at all of its points, then E is d-rectifiable.