## An expository paper on the Favard length problem

2 12 2012

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.

## A Knapp example for Salem sets on the line

26 11 2012

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. Read the rest of this entry »

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

9 03 2012

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.

## Summer school

17 02 2012

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.

## Buffon’s needles and other creatures

6 09 2011

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.)

29 08 2011

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?

## An update on differentiation theorems

7 07 2009

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.

## Maximal estimates and differentiation theorems for sparse sets

31 05 2009

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. $x\in {\mathbb R}.$

## Bourgain’s circular maximal theorem: an exposition

23 05 2009

The following spherical maximal theorem was proved by E.M. Stein in the 1970s in dimensions 3 and higher, and by Bourgain in the 1980s in dimension 2.

Theorem 1. Define the spherical maximal operator in ${\mathbb R}^d$ by

$M f(x)=\sup_{t>0}\int_{S^{d-1}}|f(x+ty)|d\sigma(y),$

where $\sigma$ is the normalized Lebesgue measure on the unit sphere $\mathbb S^{d-1}$. Then

$\| M f(x) \|_{p} \leq C\| f \|_{p}$ for all $p > \frac{d}{d-1}.$

The purpose of this post is to explain some of the main ideas behind Bourgain’s proof. It’s a beautiful geometric argument that deserves to be well known; I will also have to refer to it when I get around to describing my recent joint work with Malabika Pramanik on density theorems. Among other things, I will try to explain why the $d=2$ case of Theorem 1 is in fact the hardest.

Note that Theorem 1 is trivial for $p=\infty$; the challenge is to prove it for some finite $p$. It is known that the range of $p$ in the theorem is the best possible, but we will not worry about optimizing it in this exposition. (Not much, anyway.)

Let’s first try to get a general idea of what kind of geometric considerations might be relevant. Fix $d=2$ and $1. For the sake of the argument, let's pretend that we are looking for a counterexample to Theorem 1, i.e. a function $f$ with $\| f \|_p$ small but $\| Mf \|_p$ large. Let's also restrict our attention for the moment to characteristic functions of sets, so that $f = {\bf 1}_\Theta$ for some set $\Theta \subset {\mathbb R}^2$. Then $\| f \|_p = | \Theta |^{1/p}$. On the other hand, let $\Omega$ be the set of all $x$ for which there exists a circle $C_x$ centered at $x$ such that a fixed proportion (say, 1/10-th) of $C_x$ is contained in $\Theta$. Then

$Mf(x) \geq .1$ for all $x \in \Omega$,

and in particular $\| Mf \|_p \geq .1 | \Omega |^{1/p}$. If we could construct examples of such sets with $|\Omega |$ fixed, but $|\Theta |$ arbitrarily small, this would contradict Theorem 1. In particular, if we could construct a set $\Theta$ of measure 0 such that for every $x \in [0,1]^2$ (or some other set of positive measure) there is a circle $C_x$ centered at $x$ and contained in $\Theta$, Theorem 1 would fail spectacularly. Thus one of the consequences of Bourgain’s circular maximal theorem is that such sets $\Theta$ can't exist. (This was also proved independently by Marstrand.)

Let’s now see if we can use this type of arguments to prove the theorem.

## Density and differentiation theorems for sparse sets

8 05 2009

Over the next couple of weeks, I will be posting short expositions of various parts of an upcoming paper by Malabika Pramanik and myself on maximal estimates associated with sparse sets in ${\mathbb R}$. I’ll start by explaining some of the questions that motivated us to do this work. We first learned about them from Nir Lev. We are grateful to him for the many conversations we had at the Fields Institute and for pointing us to references that would otherwise be very hard to find.

The following question was raised and investigated by Vincenzo Aversa and David Preiss in the 1980s and 90s: to what extent can the Lebesgue density theorem be viewed as “canonical” in ${\mathbb R}$, in the sense that any other density theorem that takes into account the affine structure of the reals must follow from the Lebesgue density theorem?

Let’s make this more precise. For the purpose of this post, we will say that family ${\cal S}$ of measurable subsets of ${\mathbb R}$ has the density property if for every measurable set $E \subset {\mathbb R}$ we have

$\lim_{S \in {\cal S}, diam ( S \cup \{ 0 \} ) \to 0 } \frac{ |(x+S) \cap E | }{ |S| } = 1$ for a.e. $x\in E$.

This is slightly different from standard terminology, but there should be no danger of confusion, as we will not use any other density properties here. We write $x+S= \{ x+y:\ y\in S \}$.

The Lebesgue density theorem states that the collection of intervals $\{ (-r,r): \ r>0 \}$ has this property. It also implies that collections such as $\{(0,r):\ r>0\}$ or $\{(\frac{r}{2},r):\ r>0\}$ have it, just because the intervals in question occupy a positive and bounded from below proportion of $(-r,r)$.

But that does not exhaust all examples. For instance, consider the family $\{ I_n \}_{n=1}^\infty$, where $I_n=( \frac{ n }{ (n+1)! } , \frac{ 1 }{ n! } )$. We have $|I_n|=\frac{1}{(n+1)!}$ and $diam ( I_n \cup \{ 0 \} )= \frac{ 1 }{ n! }$, hence the Lebesgue argument no longer works. Nonetheless, this collection does have the density property, by the hearts density theorem of Preiss and Aversa-Preiss.

Note, however, that the collection in the last example is not closed under scaling $x \to r x$, $r>0$. Aversa and Preiss have in fact proved that if a family of intervals is invariant under such scaling and has the density property, then its density property must follow from the Lebesgue theorem in the manner described above.

On the other hand, if we consider more general sets than intervals, then it turns out that there are indeed scaling-invariant density theorems that are independent of the Lebesgue theorem. This was announced by Aversa and Preiss in 1987; the proof (via a probabilistic construction) was published in a 1995 preprint.

Theorem 1 (Aversa-Preiss): There is a sequence $\{ S_n \}$ of compact sets of positive measure such that $|S_n|\to 0$ and:

• $0$ is a Lebesgue density point for ${\mathbb R } \setminus \bigcup S_n$, and in particular we have $\lim_{ n\to\infty } \frac{ |S_n| }{ diam (S_n \cup \{ 0 \} ) }=0;$

• the family $\{rS_n:\ r>0, n\in {\mathbb N} \}$ has the density property.

The analogous question for $L^p$ differentiation theorems turned out to be much more difficult.

We will say that ${\cal S}$ differentiates $L^p_{loc} ( {\mathbb R} )$ for some $1\leq p\leq\infty$ if for every $f\in L^p_{loc} ( {\mathbb R} )$ we have

$\lim_{ S\in {\cal S}, diam (S\cup \{ 0 \} )\to 0 } \frac{ 1 }{ |S| } \int_{x+S} f( y ) dy = f(x)$ for a.e. $x\in E$.

For example, the Lebesgue differentiation theorem states that the collection $\{ (-r,r): r>0\}$ differentiates $L^1_{loc}( {\mathbb R })$.

The differentiation property is formally stronger than the density property, by letting $f$ range over characteristic functions of measurable sets. However, there is no automatic implication in the other direction.

The following theorem was conjectured by Aversa and Preiss in 1995, and proved very recently by Malabika Pramanik and myself (paper in preparation).

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. $x\in {\mathbb R}.$

Our construction of $S_n$, like that of Aversa and Preiss, is probabilistic. We prove that the sequence $S_n$ can be chosen so that the maximal operator associated with it is bounded on appropriate $L^p$ spaces. This in particular implies the differentiation theorem.

The exact statement of the maximal estimate, and some of the ideas from the proof, will follow in the next installment.