Category Archives: mathematics: research

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

Advertisements

2 Comments

Filed under mathematics: research

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.

2 Comments

Filed under mathematics: research

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.

If you are interested in participating, please contact Christoph Thiele (thiele@math.ucla.edu) by March 15.

Comments Off on Summer school

Filed under mathematics: research

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.)

1 Comment

Filed under mathematics: research

A question about terminology

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?

22 Comments

Filed under mathematics: research

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.

Comments Off on An update on differentiation theorems

Filed under mathematics: research

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<t<2. 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}.

Continue reading

1 Comment

Filed under mathematics: research