Vanishing sums of roots of unity and the Favard length of self-similar product sets

Vanishing sums of roots of unity and the Favard length of self-similar product sets, Discrete Analysis 2022:19, 31 pp.

An important theme in geometric measure theory is the typical size of a set when it is randomly projected. For example, suppose that $A$ is a subset of the plane with finite and non-zero 1-dimensional Hausdorff measure. 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. 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.

A well-known theorem of Besicovitch states that these two examples are typical in the following sense. 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.

The Favard length of a set $A$ is defined to be the average measure of all its projections. So Besicovitch’s result tells us that the Favard length of a “dust-like” set of finite 1-dimensional Hausdorff measure is zero. However, that leaves more precise quantitative questions unanswered. 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 $C_{n-1}$. 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. 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 $C_n$.

A 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. To build a 2-dimensional Cantor set, one can take a finite set $A\times B$, replace each point by a copy of $A\times B$ shrunk by a factor $|A||B|$, obtaining a new set $A_1\times B_1$. Then one can replace each point by a copy of $A\times B$ shrunk by a factor $|A|^2|B|^2$, and so on. The limiting set (defined in a natural way) is the Cantor set derived from $A$ and $B$. The relationship between this and sums of roots of unity arises because the authors find themselves considering the polynomials $A(x)=\sum_{a\in A}x^a$ and $B(x)=\sum_{b\in B}x^b$, and needing to know about when such polynomials vanish on the unit circle.

Here one should be thinking of $A$ and $B$ as having a fixed size, but possibly with very large elements, so the resulting polynomials, sometimes known as fewnomials, have very few terms but can have extremely high degree. Now let $z=e^{i\theta}$ be a root of $A(x)$. If $\theta$ is an irrational multiple of $\pi$, then $z$ can be dealt with using a theorem of Baker. Otherwise, if the order of $z$ has a factor in common with $|A||B|$, then it is considered to be “good”, whereas if it is coprime to $|A||B|$ then it is bad. One can then factorize $A(x)$ into $A_g(x)A_b(x)$ – the good and bad parts. The influence of the good roots on the estimates can be gauged using a set called the “structured set of small values” of $A_g$ on the unit circle.

It remains to understand the influence of the bad roots. Here it turns out that one needs to find “structured sets of large values”, which are quite large subsets $\Gamma$ of the unit circle such that $\Gamma\Gamma^{-1}$ is far from all the bad roots. And in order to find such sets, one ends up considering vanishing sums of roots of unity, a subject that has received plenty of attention quite independently of this problem.

A major result concerning such sums is a theorem of Lam and Leung, which states that if $z_1,\dots,z_k$ are $N$th roots of unity that sum to zero, then $k$ is at least as big as the smallest prime divisor of $N$. Obviously this bound is sharp, in general, but this paper shows that under certain additional assumptions it can be improved. The authors go on to use this improvement to extend what is known about the Favard length of rational Cantor sets. In particular, where Bond, Łaba and Volberg were able to obtain good estimates (for the decay of the Favard length as the construction of the Cantor set proceeds) when $A$ and $B$ have size at most 6 and the polynomials $A(x)$ and $B(x)$ have at most one cyclotomic divisor, this paper works for arbitrary sets $A$ and $B$ with sizes up to 10 (and could in principle go beyond that, but that would involve a case analysis that rapidly becomes unwieldy).

The paper combines algebra and analysis to
improve our understanding of a very tough question in geometric measure theory. The tools developed by the authors are likely to have several further uses in the area.

The following video is not specifically about the results of this paper, but it includes a discussion them as part of a broader discussion of this subarea of geometric measure theory.