# Category Archives: mathematics: research

## Finite configurations in sparse sets

One more paper finished: “Finite configurations in sparse sets,” joint with Vincent Chan and Malabika Pramanik. The paper is available here, and here is the arXiv link.

Very briefly, the question we consider is the following. Let $E \subseteq \mathbb{R}^n$ be a closed set of Hausdorff dimension $\alpha$. Given a system of $n \times (m-n)$ matrices $B_1, ... ,B_k$ for some $m \geq n$, must $E$ contain a “non-trivial” k-point configuration

$(1)\ \ \ x + B_1 y,\ ...,\ x + B_k y$

for some $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^{m-n}$?

In general, the answer is no, even when $\alpha=n$. For instance, Keleti has constructed 1-dimensional subsets of $\mathbb{R}$ that do not contain a similar copy of any given triple of points $(x,y,z)$ (in fact, his construction can avoid all similar copies of all such triples from a given sequence), as well as 1-dimensional subsets of $\mathbb{R}$ that do not contain any non-trivial “parallelograms” $\{x, x+y, x+x, x+y+z\}$. In $\mathbb{R}^2$, given any three distinct points $a,b,c$, Maga has constructed examples of sets of dimension 2 that do not contain any similar copy of the triangle $abc$; he also constructed sets of full dimension in $\mathbb{R}^n$, for any $n\geq 2$, 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 $E\subset \mathbb{R}$ has dimension close enough to 1, and if it also supports a measure obeying appropriate dimensionality and Fourier decay estimates, then $E$ must contain a non-trivial 3-term arithmetic progression. The same proof applies to any other configuration $x,y,z$, 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 $B_j$, 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 $B_j$, if $E\subset \mathbb{R}^n$ has dimension close enough to $n$, and if it supports a measure with dimensionality and Fourier decay conditions similar to those in my paper with Pramanik, then $E$ 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 $B_j$ 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 $\mathbb{R}^2$. Let $a,b,c$ be three distinct points in the plane. Suppose that $E\subset\mathbb{R}^2$ satisfies the assumptions of our main theorem. Then $E$ must contain three distinct points $x,y,z$ such that the triangle $\triangle xyz$ is a similar (possibly rotated) copy of the triangle $\triangle abc$.
• Colinear triples in $\mathbb{R}^n$. Let $a,b,c$ be three distinct colinear points in $\mathbb{R}^n$. Assume that $E\subset\mathbb{R}^n$ satisfies the assumptions of our main theorem. Then $E$ must contain three distinct points $x,y,z$ that form a similar image of the triple $a,b,c$.
• Parallelograms in $\mathbb{R}^n$. Assume that $E\subset\mathbb{R}^n$ satisfies the assumptions of our main theorem. Then $E$ contains a parallelogram $\{x,x+y,x+z,x+y+z\}$, where the four points are all distinct.

Filed under mathematics: research

## Visibility of unrectifiable planar sets

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 $E$ in the plane, and a point $a$ not in $E$. Define $P_a$ to be the radial projection from $a$:

$P_a(x) = \frac{x-a}{|x-a|}$

Then $P_a(E)$ is the set of angles at which $E$ is visible from $a$. Our “visibility problem” is then to estimate the size of $|P_a(E)|$, or equivalently, the proportion of the part of the field of vision that $E$ takes up for an observer situated at $a$.

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 $K_n$ for the $n$-th iteration of this set, and $K$ for the Cantor set $K = \bigcap_{n=1}^\infty K_n$.

What can we say about the visibility of $K$ from points $a$ in the plane? We will assume that $a \notin K$, 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 $P_a(K)$, as expressed in terms of its Lebesgue measure and/or Hausdorff dimension.

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## An expository paper on the Favard length problem

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.

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## A Knapp example for Salem sets on the line

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 $L^2$ densities on the sphere, and on the other hand, Stein’s restriction conjecture for $L^\infty$ 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

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## Buffon’s needles: a very short expository note

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.

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## Summer school

Christoph Thiele, Malabika Pramanik and myself are organizing the summer school “Harmonic Analysis, Geometric Measure Theory and Additive Combinatorics”.

The school will focus on questions at the interface of harmonic analysis, geometric measure theory and additive combinatorics, including applications of harmonic analytic and additive combinatorial methods (restriction theorems, trigonometric polynomial estimates) to measure theoretic problems. In particular, many questions in geometric measure theory and harmonic analysis (for example, concerning projections of sets or the occurrence of prescribed patterns in them) explore various concepts of “randomness” of sets. Similar phenomena for discrete sets have been investigated in additive combinatorics, and we hope that the methods developed there can be transferred to a continuous setting. The school will introduce the participants to a selection of problems of this type. A basic familiarity with measure theory, harmonic analysis and probability theory will be expected.

The school will take place from June 24 – 29. 2012 at Catalina Canyon Resort on Catalina Island, California. Space is limited to 12 participants, normally graduate students or postdocs. Participants are required to prepare a lecture on a topic chosen from our list and submit a short written summary of it prior to the school.

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## Buffon’s needles and other creatures

It took several false starts, complete changes of direction and various other mishaps, but “Buffon’s needle estimates for rational product Cantor sets” (aka Project Lamprey), joint with Matthew Bond and Alexander Volberg, has been completed and posted on the arXiv. I will add the link as soon as it goes live; in the meantime, you can also download the paper from my web page.

(Updated: the arXiv link is here, and here is the revised version with minor corrections and clarifications.)

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There’s a project that I and collaborators have been working on for a fairly long time now. It is almost finished, at least the first stage of it, and I will have more to say about it once we have posted the paper on the arXiv. In the meantime, though, there is a very important question that we need to consider.

Would it be well received in the community if we referred to a certain class of sets appearing in the paper as “non-parasitic lampreys”?

For all we know, the community does not currently harbour any particular feelings towards such sets. They have come up in a couple of places over the years, but their possible parasitic behaviour has not really been investigated until now. We can prove that certain particular lampreys of interest are indeed non-parasitic, which is good for us. By way of contrast, “eels” are somewhat more straightforward than lampreys. That makes them easy to manage when they’re small, but otherwise they’re still troublemakers.

This would be a radical departure from the naming conventions established in, say, physics or algebraic geometry. While those of course abound in colourful vocabulary, much of it refers to various forms of enchantment, awe, amazement, pleasure and wonder, not necessarily the feelings that lampreys tend to inspire. But… our unofficial terminology fits so nicely, I’d be quite reluctant to part with it.

What do you think?

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## An update on differentiation theorems

Malabika Pramanik and I have just uploaded to the arXiv the revised version of our paper on differentiation theorems. The new version is also available from my web page.

Here’s what happened. In the first version, we proved our restricted maximal estimates (with the dilation parameter restricted to a single scale) for all $p>1$; unfortunately our scaling analysis worked only for $p\geq 2$, therefore our unrestricted maximal estimates and differentiation theorems were only valid in that range. However, just a few days after we posted the paper, Andreas Seeger sent us a “bootstrapping” scaling argument that works for $p$ between 1 and 2. With Andreas’s kind permission, this is now included in the new version. The updated maximal theorem is as follows.

Theorem 1. There is a decreasing sequence of sets $S_k \subseteq [1,2]$ with the following properties:

• each $S_k$ is a disjoint union of finitely many intervals,
• $|S_k| \searrow 0$ as $k \rightarrow \infty$,
• the densities $\phi_k=\mathbf 1_{S_k}/|S_k|$ converge to a weak limit $\mu$,
• the maximal operators

${\mathcal M} f(x):=\sup_{t>0, k\geq 1} \frac{1}{|S_k|} \int_{S_k} |f(x+ty)|dy$

and

${\mathfrak M} f(x) = \sup_{t > 0} \int \left| f(x + ty) \right| d\mu(y)$

are bounded on $L^p({\mathbb R})$ for $p >1$.

Our differentiation theorem has been adjusted accordingly.

Theorem 2. Let $S_k$ and $\mu$ be given by Theorem 1. Then the family ${\cal S} =\{ rS_k:\ r>0, n=1,2,\dots \}$ differentiates $L^p( {\mathbb R})$ for all $p>1$, in the sense that for every $f \in L^p$ we have

$\lim_{r\to 0} \sup_{n} \frac{ 1 }{ r|S_n| } \int_{ x+rS_n } f(y)dy = f(x)$ for a.e. $x\in {\mathbb R}.$

Furthermore,

$\lim_{r\to 0} \int f(x+ry) d \mu (y) =f(x)$ for a.e. $x\in {\mathbb R}.$

What about $p=1$? I had the good luck of meeting David Preiss in Barcelona right after Malabika and I had finished the first version of the preprint. I explained our work; we also spent some time speculating on whether such results could be true in $L^1$. Next day, David sent me a short proof that our Theorem 2 cannot hold with $p=1$ for any singular measure $\mu$ supported away from 0. (The same goes for sequences of sets $S_k$ as above, by a slight modification of his argument.) We are grateful to David for letting us include his proof in the new version of our paper.

We have also polished up the exposition, fixed up the typos and minor errors, etc. One other thing we have added (to the arXiv preprint – we are not including this in the version we are submitting for publication) is a short section on how to modify our construction of $S_k$ so that the limiting set $S$ would also be a Salem set. The argument is very similar to the construction in our earlier paper on arithmetic progressions, so we only sketch it very briefly.

I’ll be on vacation throughout the rest of July. I’ll continue to show up here on this blog – I might actually write here more often – and I’ll finish up a couple of minor commitments, but you should not expect any more serious mathematics from me in the next few weeks.

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## Maximal estimates and differentiation theorems for sparse sets

Malabika Pramanik and I have just uploaded to the arXiv our paper Maximal operators and differentiation theorems for sparse sets. You can also download the PDF file from my web page.

The main result is as follows.

Theorem 1. There is a decreasing sequence of sets $S_k \subseteq [1,2]$ with the following properties:

• each $S_k$ is a disjoint union of finitely many intervals,
• $|S_k| \searrow 0$ as $k \rightarrow \infty$,
• the densities $\phi_k=\mathbf 1_{S_k}/|S_k|$ converge to a weak limit $\mu$,
• the maximal operators

${\mathcal M} f(x):=\sup_{t>0, k\geq 1} \frac{1}{|S_k|} \int_{S_k} |f(x+ty)|dy$

and

${\mathfrak M} f(x) = \sup_{t > 0} \int \left| f(x + ty) \right| d\mu(y)$

are bounded on $L^p({\mathbb R})$ for $p\geq 2$.

It turns out that the set $S=\bigcup_{k=1}^\infty S_k$ does not even have to have Hausdorff dimension 1 – our current methods allow us to construct $S_k$ so that $S$ can have any dimension greater than 2/3. We also have $L^p\to L^q$ estimates as well as improvements in the range of exponents for the “restricted” maximal operators with $1. See the preprint for details.

Theorem 1 allows us to prove a differentiation theorem for sparse sets, conjectured by Aversa and Preiss in the 1990s (see this post for a longer discussion).

Theorem 2. There is a sequence $[1,2]\supset S_1\supset S_2\supset\dots$ of compact sets of positive measure with $|S_n| \to 0$ such that ${\cal S} =\{ rS_n:\ r>0, n=1,2,\dots \}$ differentiates $L^2( {\mathbb R})$. More explicitly, for every $f \in L^2$ we have

$\lim_{r\to 0} \sup_{n} \frac{ 1 }{ r|S_n| } \int_{ x+rS_n } f(y)dy = f(x)$ for a.e. $x\in {\mathbb R}.$

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