The relationship between this and sums of roots of unity arises because the authors find themselves considering the polynomials A ( x ) = ∑ a ∈ A x a and B ( x ) = ∑ b ∈ B x b, and needing to know about when such polynomials vanish on the unit circle. The limiting set (defined in a natural way) is the Cantor set derived from A and B. Then one can replace each point by a copy of A × B shrunk by a factor | A | 2 | B | 2, and so on. To build a 2-dimensional Cantor set, one can take a finite set A × B, replace each point by a copy of A × B shrunk by a factor | A | | B |, obtaining a new set A 1 × B 1. Ī paper of Bond, Łaba and Volberg from 2014 relates the question (for Cantor sets defined using rational parameters) to a question about sums of roots of unity that equal zero. A nice way of looking at this question is to think of it as asking what the probability is that “Buffon’s needle” – that is, a needle dropped randomly in the vicinity of C n (any sensible definition will do here) – intersects C n. We may know that the limit of the Favard lengths of the C n is zero, but that does not tell us anything about the speed at which the lengths decrease, which turns out to be a surprisingly difficult problem. For instance, a Cantor set C is usually presented as the intersection of nested sets C n, where C n is made up of a fixed number of copies of C n − 1. However, that leaves more precise quantitative questions unanswered. So Besicovitch’s result tells us that the Favard length of a “dust-like” set of finite 1-dimensional Hausdorff measure is zero. The Favard length of a set A is defined to be the average measure of all its projections. ![]() Either some portion of A of positive 1-dimensional Hausdorff measure is contained in a rectifiable curve, in which case almost all projections have positive measure, or that is not the case (in which case A is called purely unrectifiable), and almost all projections have measure zero. But there are other examples with different behaviour: for example, if A is a product of two Cantor sets, defined with parameters that are chosen to ensure that A has Hausdorff dimension 1, then it turns out that with probability 1 the measure of P ( A ) is zero.Ī well-known theorem of Besicovitch states that these two examples are typical in the following sense. If we choose a random line L through the origin and let P be the orthogonal projection to L, what can we say about the size of P ( A ) ? If A is a line segment, we see that with probability 1 the projection P ( A ) has non-zero measure. ![]() For example, suppose that A is a subset of the plane with finite and non-zero 1-dimensional Hausdorff measure. Vanishing sums of roots of unity and the Favard length of self-similar product sets, Discrete Analysis 2022:19, 31 pp.Īn important theme in geometric measure theory is the typical size of a set when it is randomly projected.
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