The Rudin (Hardy-Littlewood) Conjecture, Notes 2: A closer look at Λ(p)-sets.

This is the second post on Rudin’s conjecture. For the first introductory notes see here. In this post I will try to build some more intuition on {\Lambda(p)}-sets by studying some examples and discussing their properties. We will also discuss some basic question that have been studied in the literature of {\Lambda(p)}-sets.

1. Lacunary sequences and {\Lambda(p)}-sets.

Let us briefly recall the definition of {\Lambda(p)}-sets. For {E\subset \mathbb Z}, let {f:\mathbb T \rightarrow {\mathbb C}} be an {E}-polynomial, that is {\hat f(n)=0} for all {n\notin E} and {f} is a trigonometric polynomial. For {0<p<\infty} we call {E} a {\Lambda(p)}-set if there exists a {q<p} such that

\displaystyle  	\|f\|_p\lesssim_p \|f\|_q, \ \ \ \ \ (1)

for all {E}-polynomials {f}. Remember that if (1) holds for some {0<q<p} then it holds for all such {q} and this is just a consequence of Hölder’s inequality.

It is pretty obvious that not every subset of the integers can be a {\Lambda(p)}-set. In particular the integers themselves are not a {\Lambda(p)}-set and this can be very easily verified by checking against the Dirichlet Kernel for which we have {\|D_N\|_p \simeq N^\frac{1}{p'}}, {p'} being the dual exponent of {1<p\leq \infty} and {\|D\|_1\simeq \log N}.

On the other hand, the easiest example of a {\Lambda(p)}-set is probably a lacunary sequence.

1.1. Hadamard lacunary sequences

Definition 1 Let {A=\{a_1,a_2,a_3\ldots\}} be a sequence of positive integers. The sequence {\{a_k\}_{k=1} ^\infty} is called lacunary in the sense of Hadamard if there exists some constant {\lambda>1} such that

\displaystyle  \frac{a_{k+1}}{a_k}>\lambda,\quad k=1,2,3,\ldots

Now it is a classical result (due to Salem and Zygmund) that a Hadamard lacunary sequence {A} is a {\Lambda(p)}-set. That is, we have

Theorem 2 (Salem, Zygmund) Let {A=\{a_1,a_2,\ldots\}} be a lacunary sequence of positive integers. Then {A} is a {\Lambda(p)}-set for all {0<p<\infty}.

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The Rudin (Hardy-Littlewood) Conjecture, Notes 1: Introduction and basic facts.

1. Introduction.

This is the first of a series of posts concerning the Rudin-Hardy-Littlewood Conjecture. To give a taste of the problem right away let us consider {f} to be a trigonometric polynomial of the form

\displaystyle  \begin{array}{rcl}  	f(\theta)=\sum_{n=1} ^N a_n e^{i n^2\theta}, \quad \theta\in \mathbb T, \end{array}

where {a_n\in\mathbb C} for {1\leq n \leq N}. The main question we are interested in is whether one has an inequality of the form:

Conjecture 1 (Rudin’s Conjecture) For all {2<p<4} we have that

\displaystyle  	\|f\|_p\lesssim _p \|f\| _2, \ \ \ \ \ (1)

where the implied constant depends only on {p}.

Conjecture 1 was stated in this form by Walter Rudin himself for example in [R] but the the first (and essentially only) results on this question go back to Hardy and Littlewood (see for example [R]).

Inequalities of the form (1) have deep number theoretic implications. Continue reading

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Μάλιστα Κύριε

Μα τις νυχτιές σα συλλογιέμαι, τα μάτια της τα μενεξιά, φοβάμαι και αναρωτιέμαι, πώς θα σ’αντέ- πώς θα σ’αντέξω, μοναξιά.

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It’s a sad and beautiful world.

Don’t have much to say at this hour. Except for this:

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The Discrepancy function in two dimensions.

Tomorrow I am traveling to Heraklion, Crete, where among other things I will give a talk on Discrepancy in two Dimensions. The talk is based on a recent paper with D. Bilyk, M. Lacey and A. Vaghasrhakyan. Here follows an outline of the talk which can also be used as an easier first reading of the paper (which is admittedly quite technical).

1. Introduction.

Everything will take place in the unit cube {Q:= [0,1]^d \subset \mathbb R^d}. For {\vec x = (x_1,\ldots,x_d)\in Q} we write

\displaystyle  \begin{array}{rcl}  	[0,\vec x) := [0,x_1)\times \cdots \times [0,x_d), \end{array}

for the rectangle anchored at {0} and at {\vec x}. Let {\mathcal P_N} be an {N}-point distribution in {[0,1]^d}:

\displaystyle  \begin{array}{rcl}  	\mathcal P_N = \{p_1,p_2,\ldots,p_N\} \subset [0,1]^d. \end{array}

Definition 1 The discrepancy function of {\mathcal P _N} is

\displaystyle  D_N(x):=\sharp \mathcal P_N\cap [0,\vec x) - N |[0,\vec x)|.

The first term in the previous definition is sometimes referred to as the counting part:

\displaystyle  \mathcal C _N(x):=\sharp \mathcal P_N\cap [0,\vec x) =\sum_{p\in\mathcal P_N} \mathbf 1 _{[\vec p,\vec 1)}(x)

Obviously this term counts the number of points of {\mathcal P_N} inside {[0,\vec x)}. We call the second term in the definition of the discrepancy function `the linear part’:

\displaystyle \mathcal L_N(x):= N|[0,\vec x)|.

Here {|E| } denotes the Lebesgue measure of a set {E}. The linear part expresses the expected number of points in {[0,\vec x)} if we pick the points uniformly and independently at random.

It turns out that the `size’ of this function (in many different senses) must necessarily grow to infinity with {N\rightarrow \infty.} Continue reading

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Le vent nous portera.

This is a naive short story. On the one hand it is mainly the result of procrastinating online on a Sunday night. On the other hand, a more positive person than me would see this as an instance of a -you can find whatever you want online with only so little initial information- sort of situation. So for reasons I will not go into, I wanted to find a particular song I used to listen to when I was living in Paris between January and July of 2004. I was listening to that song a lot on my small radio, on a radio station called OUI FM; arguably a ridiculous name for a radio station but the music was quite good according to my taste. Anyways, I was in Paris, and the particular radio station played a lot of indie songs, both french and otherwise. So, I was particularly fond of a french song that was obviously popular in France at the time. However, I never knew the name of the song, or the name of the band that sang the song, or well, even if I knew back then, I had absolutely no recollection a couple of minutes ago.

Usually when you have some initial piece of information, it’s just a matter of time to recover whatever it is that you wish to know, online. I’m not of course talking about knowing the name of the band, or the name of the song. But suppose, for example, that you remember a non-trivial part of the lyrics, meaning a part that could more or less uniquely define the song. Then Google the lyrics you remember, sort out the irrelevant results, and you are done. I believe this algorithm is largely successful even when trying to recover more obscure information that’s not so well documented online as music is. On the other hand, the initial data you need is not that trivial.

This time I had very little to go with. I remembered the song in the sense that I could recognize it if I listened to it. And I remembered one word from the lyrics: ‘vent’. Not so much to go with. I mean the first step is to Google the words vent, french or français, indie, pop, 2003/2004 and see what you get. I didn’t get anywhere really with this first naive approach. Trying at more music-specialized web sites only gave me some songs that I also listened to that radio station back then. It was not what I was looking for but now I had one more piece of information to go with. Similar artists! I remembered the similar artists section at last.fm. After that the story gets boring. I finally managed to narrow down the candidates,  played a couple of songs in preview mode et voilà!

OK, probably not much for a Sunday night. But, you feel kind of satisfied when eventually this sort of pursuit goes through. In my opinion, this is a non-trivial example of internet search. Imagine that instead of a song you absolutely need to find out if a certain theorem is known to be true. Not that this necessarily covers a more substantial need. In any case, I am impressed that you can begin with such vague information, bootstrap it if necessary, and come up with a concrete result. In less than 30 minutes…

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Some first thoughts on starting a blog.

I am starting this blog motivated by several, seemingly independent, reasons. Probably the most important among them is trying to keep up with a general trend that I certainly feel that concerns me; the Internet is rapidly changing from being a basic static tool for Mathematicians to the place where Mathematics actually happen. Of course, confining this discussion to Mathematics is quite narrow-minded. This is a much more general phenomenon in science, art, in fact, in anything that matters to people. But I will stick to Mathematics because that is this is the subject I am more informed and more involved in. In all the other areas I am just another user who benefits everyday from the available online information, much like any interested person in Mathematics could look up a definition at Wikipedia. I will look up lyrics to songs I like or find information about a movie I would like to watch. I enjoy all the geeky developments happening, I am experimenting with Google Wave and I use many other tools like Google Maps and Google Search (really who doesn’t?) But I am not dynamically part of this; I live on the user end. In Mathematics I could say that I live on both ends. Or rather, that my intention starting this blog, is to experiment a bit more on being part of the dynamic development of the Internet with respect to Mathematics.

Another, more practical reason for starting this blog, is that it will serve as a companion site for future seminars I will organize or participate in, as well as courses that at some point in my career I will hopefully give. So, in a way, I’m using the general public in order to gain experience and feedback on how a course or a seminar should be organized and supported online. That way  the experimenting part on the actual students to come will be somewhat reduced. I admit that my original intention was to use Google Wave (GW) for this task which seems to provide a more user-friendly and less geeky environment to carry such projects through. However, GW is very far from being stable and it is also very far from being practical from a mathematical point of view. The latex support is pretty elementary and still under basic development plus the whole platform is in preview mode which means that not everyone can use it and so on. On the other hand, WordPress has proved to be a very efficient platform for Mathematical blogs, and maybe the “geekiness” involved is just necessary.

Finally, I am a typical obsessive compulsive person. When I am reading an article or a proof in a book, I really need to write down all the details and count all the \epsilon‘s in order to feel I understand what’s going on. Usually I do that on some scrap paper which then gets lost, thrown away, recycled, whatever. Many times I waste a significant amount of time trying to reproduce a specific argument that I had figured out at a previous moment. In other words, I hope that this blog will serve as my own reference as well. Hopefully it can help as such to other people besides me. I think I will focus mostly on review posts, analyzing a certain subject I feel I understand quite well. Besides specific mathematical topics (mostly in Harmonic Analysis) I plan to use this blog in order to discuss issues which are not of mathematical interest, at least not directly so. This could include music, web applications, books, movies, but also subjects that are tangentially related to mathematics as latex related issues, job seeking issues, as well as the discussion I alluded to earlier, that is how mathematics happen and evolve online. Hopefully, I will soon start a post addressing exactly this issue with several links to web sites, blogs and wiki’s that I am aware of.
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