Since a number of people asked, here are the slides from my ICM talk yesterday. I have also posted them on my preprints page. I believe the talk was recorded and the video will presumably be available from the ICM webpage. Alternatively, you can read my ICM proceedings paper for a longer version.
Category Archives: mathematics: research
Somewhat belatedly, here’s the expository paper I wrote for the ICM Proceedings: a short overview of my work with Malabika Pramanik, Vincent Chan and Kyle Hambrook on harmonic analytic estimates for singular measures supported on fractal sets.
The connection between Fourier-analytic properties of measures and geometric characteristics of their supports has long been a major theme in Euclidean harmonic analysis. This includes classic estimates on singular and oscillatory integrals associated with surface measures on manifolds, with ranges of exponents depending on geometric issues such as dimension, smoothness and curvature.
In the last few years, much of my research has focused on developing a similar theory for fractal measures supported on sets of possibly non-integer dimension. This includes the case of ambient dimension 1, where there are no non-trivial lower-dimensional submanifolds but many interesting fractal sets. The common thread running through this work is that, from the point of view of harmonic analysis, “randomness” for fractals is often a useful analogue of curvature for manifolds. Thus, “random” fractals (constructed through partially randomized procedures) tend to behave like curved manifolds such as spheres, whereas fractals exhibiting arithmetic structure (for instance, the middle-thirds Cantor set) behave like flat surfaces. There is a clear connection, at least on the level of ideas if not specific results, to additive combinatorics, where various notions of “randomness” and “arithmetic structure” in sets of integers play a key role.
The paper discusses three specific questions that I have worked on: restriction estimates, differentiation estimates, and Szemeredi-type results. I’ve also mentioned some open problems. At this point, I feel like we’re only started to scratch the surface here; there is much more left to do, for example optimizing the exponents in some of the estimates I’ve mentioned and, perhaps more importantly, figuring out what properties of fractal measures determine such exponents.
Very briefly, the question we consider is the following. Let be a closed set of Hausdorff dimension . Given a system of matrices for some $m \geq n$, must contain a “non-trivial” k-point configuration
for some and ?
In general, the answer is no, even when . For instance, Keleti has constructed 1-dimensional subsets of that do not contain a similar copy of any given triple of points (in fact, his construction can avoid all similar copies of all such triples from a given sequence), as well as 1-dimensional subsets of that do not contain any non-trivial “parallelograms” . In , given any three distinct points , Maga has constructed examples of sets of dimension 2 that do not contain any similar copy of the triangle ; he also constructed sets of full dimension in , for any , that do not contain non-trivial parallelograms.
Additive combinatorics suggests, however, that sets that are “random” in an appropriate sense should he better behaved in that regard. Along these lines, Malabika Pramanik and I proved in an earlier paper that if has dimension close enough to 1, and if it also supports a measure obeying appropriate dimensionality and Fourier decay estimates, then must contain a non-trivial 3-term arithmetic progression. The same proof applies to any other configuration , with the dimension bound depending on the choice of configuration.
This paper gives a multidimensional analogue of that result. We define, via conditions on the matrices , a class of configurations that can be controlled by Fourier-analytic estimates. (Roughly, they must have enough degrees of freedom, and they must be “non-degenerate” in an appropriate sense.) For such , if has dimension close enough to , and if it supports a measure with dimensionality and Fourier decay conditions similar to those in my paper with Pramanik, then must indeed contain a non-trivial configuration as in (1).
The main new difficulty is dealing with the complicated geometry of the problem. There’s a lot of linear algebra, multiple coordinate systems, multilinear forms, and a lot of estimates on integrals where the integrand decays at different rates in different directions. At one point, we were actually using a partition of unity similar to those I remembered from my work in multiparticle scattering theory a very long time ago. That didn’t make it into the final version, though – we found a better way.
I won’t try to state the conditions on here – they’re somewhat complicated and you’ll have to download the paper for that – but I’ll mention a few special cases of our theorem.
- Triangles in . Let be three distinct points in the plane. Suppose that satisfies the assumptions of our main theorem. Then must contain three distinct points such that the triangle is a similar (possibly rotated) copy of the triangle .
- Colinear triples in . Let be three distinct colinear points in . Assume that satisfies the assumptions of our main theorem. Then must contain three distinct points that form a similar image of the triple .
- Parallelograms in . Assume that satisfies the assumptions of our main theorem. Then contains a parallelogram , where the four points are all distinct.
Matt Bond, Josh Zahl and I have just completed a new paper “Quantitative visibility estimates for unrectifiable sets in the plane,” now available on the arXiv. This post is an informal introduction to the paper; for more details, you will need to download the actual article.
There are several questions known as “visibility problems”, and the one we address is the following. We are given a compact set in the plane, and a point not in . Define to be the radial projection from :
Then is the set of angles at which is visible from . Our “visibility problem” is then to estimate the size of , or equivalently, the proportion of the part of the field of vision that takes up for an observer situated at .
One class of sets that we will study is 1-dimensional unrectifiable self-similar sets. A good example to keep in mind is the “4-corner set,” constructed via a Cantor iteration as follows. Start with a square, divide in into 16 congruent squares, and keep the 4 small squares at the corners, discarding the rest. Repeat the same procedure for each of the 4 small surviving squares, then iterate the construction. The first and second stage of the iteration are shown below.
We will use for the -th iteration of this set, and for the Cantor set .
What can we say about the visibility of from points in the plane? We will assume that , so as to avoid trivial debates over whether a point is visible from itself. We will be asking this question in terms of the size of , as expressed in terms of its Lebesgue measure and/or Hausdorff dimension.
In case there’s anyone here who’s interested in the Favard length problem, I have just finished an expository paper written for the proceedings of the 2012 Abel Symposium. There has been a good deal recent progress on the subject in recent years, starting with this 2008 paper by Nazarov, Peres and Volberg, through my paper with Zhai, two papers by Bond and Volberg, and most recently my paper with Bond and Volberg. This exposition focuses especially on the number-theoretic aspects of the question for rational product sets, developed mostly in the BLV paper, although some of it goes back to the earlier papers. You can think of it as BLV-lite if you wish.
I have tried to keep the exposition as simple as possible, omitting many of the technicalities and focusing on examples where we can deal with just one number-theoretic issue at a time (as opposed to BLV, where we must combine the different methods together). I’ve also added a good deal of discussion and commentary. This makes the paper a bit more verbose than what I’m used to, but most of this was written in response to questions that I have actually been asked, so I hope that this will be a useful companion paper to BLV and the other references. Also, I did have a deadline for this, so a couple of things (notably the “Poisson lemma” in Section 3.1) got short shrift, and I probably would have found a few more typos and other such if I’d had more time to chase them. Oh well.
There are a couple of new things at the end of the paper. One is Conjecture 4.6. Matt Bond and I came up with this while trying to figure out whether the assumption on the cardinalities of product sets in BLV can be dropped. If the conjecture turns out to be true, than we can indeed drop that assumption. We have some supporting evidence for various special cases, but we don’t know how to prove it in general.
The second part that has not been published previously concerns “random 4-corner sets”. Peres and Solomyak (Pacific J. Math. 2002) proved that for a randomized version of the 4-corner set construction, the expected Favard length asymptotics is in fact C/n. This is a very nice geometric argument, but I found the original proof quite hard to read, so I reworked and simplified it some time ago. This is included here in Section 5.
The paper can be downloaded here. It will also be posted on the arXiv, if I can figure out how to post LATEX files with pictures.
The restriction phenomenon in harmonic analysis is best known for surface measures on manifolds. A classical example is the unit sphere, where on the one hand we have the Stein-Tomas restriction theorem for densities on the sphere, and on the other hand, Stein’s restriction conjecture for densities remains open. (Partial intermediate results are also available, but that is a longer story that will have to wait for another time.)
However, restriction estimates can also be proved for fractal sets. Continue reading
One of the things I have been working on in the last few years is the Favard length problem. The question is to estimate the average length of a 1-dimensional projection of (a finite iteration of) a 1-dimensional self-similar Cantor set in the plane. My work with Zhai, and especially with Bond and Volberg, has pointed to connections with classical questions in number theory, including tilings of the integers, diophantine approximation of logarithms of algebraic numbers, and vanishing sums of roots of unity.
If you would like to find out a little bit more about this, but don’t necessarily feel like reading long technical papers that rely on several other long technical papers, then this very short expository note (3 pages plus short bibliography) might be for you. It was written for the CMS Notes and has about the length they require. I found myself wanting to make it longer, if only to include more references to the history and context of the problem. (I never even mentioned Comte de Buffon.) I might write a more substantial expository paper when I have the time.