À la…

A few months ago, I wondered what could be the largest cluster of foreign words in the Oxford English Dictionary, citing the examples of femme something-or-other and sympathique and company. It turns out that there is a much larger one! Here is the à la cluster:

à la 1579
à la bonne heure 1750
à la broche 1806
à la brochette 1821
à la carte 1816
à la crème 1741
à la débandade 1779
à la fourchette 1817
à la Française 1589
à la modality 1753
à la mode 1637
à-la-modeness 1669
à la mort 1536
à la page 1930
à la roi 1852
à la royale 1853
à la Russe 1775
à la Turquie 1676

That’s no less than 18 items (the date on the right is the first OED citation). It’s interesting that so many have to do with food, and even more that three or four are basically synonyms of “in fashion” (this is what à la page basically means). I have to admit to being partial to à-la-modeness for its translanguage qualities, although I don’t know if I will be able to use it intelligently anytime soon (though one never knows; after all, I did manage to sneak ptarmigan in a recent paper…)

Bounded gaps between primes!

And so it came to pass, that an almost millenial quest found a safe resting place…

Like all analytic number theorists, I’ve been amazed to learn that Yitang Zhang has proved that there exist infinitely many pairs of prime numbers $latex \ell<p$ with $latex p-\ell$ bounded by an absolute constant $latex C$.

So, how did he do it?

Well, since the paper just became available, I don’t have anything intelligent to say yet on the new ideas that he introduced (but I certainly hope to come back to this!). However, one can easily list those previously-known tools that he uses, which involve some of the deepest and most clever results in analytic number theory of the last 30 to 35 years.

(1) At the core, the proof is based on the method discovered about ten years ago by Goldston, Pintz and Yıldırım to show that

$latex \liminf \frac{p_{n+1}-p_n}{\log n}=0.$

As I discussed a while back, this remarkable result — besides its intrinsic interest — was notable for being the first to bring the problem of bounded gaps between primes within a circle of well-studied and widely believed conjectures on primes in arithmetic progressions to large moduli. Precisely, Goldston, Pintz and Yıldırım had derived the statement above, after many ingenious steps, by applying the Bombieri-Vinogradov Theorem, and they showed that any progress beyond it towards the so-called Elliott-Halberstam Conjecture would imply the bounded gap property. However, in my former blog post, I discussed why it seemed extremely difficult to go in that direction…

(2) … despite the existence of some results going beyond the Bombieri-Vinogradov theorem, due first to Fouvry-Iwaniec and later improved by Bombieri-Friedlander-Iwaniec; but Zhang uses indeed some of the ideas behind these results…

(3) … results which themselves depend crucially on two big ideas: the well-factorable weights of the linear sieve, due to Iwaniec, and the development and applications of the Kuznetsov formula and other results concerning the spectral theory of automorphic forms and estimates for sums of Kloosterman sums, the outcome of the work of Deshouillers and Iwaniec (actually, at first glance, it seems that Zhang does not explicitly use those results arising from the Kuznetsov formula; he does reach sums with incomplete Kloosterman sums which the spectral methods are designed to handle, but he can deal with them with the Weil bound only; this might be a place where the result can be improved…)

(4) … but furthermore, Zhang uses also an estimate for a certain character sum over finite fields which had appeared in the work of Friedlander and Iwaniec on the exponent of distribution for the ternary divisor function; this sum is a three-variable additive character sum, and its estimation (with square-root cancellation), proved by Bombieri and Birch in an Appendix to the paper of Friedlander and Iwaniec, depends crucially on the Riemann Hypothesis over finite fields of Deligne.

Here are some references to surveys or explanations of some of these tools. Amusingly, I have written something on most of them…

  • There have been many surveys of the work of Goldston, Pintz and Yıldırım, and in particular I wrote a Bourbaki report on it, which may be interesting to those who read French;
  • Concerning the automorphic Kloostermania that comes into the Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec circle of ideas (although it is apparently not needed for Zhang’s proof…), I happened to write a few years ago, for a book on Poincaré’s mathematical work, an account of the applications of Poincaré series to analytic number theory, which are used to prove the Kuznetsov formula;
  • Fouvry has written a survey Cinquante ans de théorie analytique des nombres from the point of view of sieve methods, which discusses the philosophy of extending the ranges of exponents of distribution for important sequences, as well as the well-factorable weights of Iwaniec;
  • Fans of trace functions may remember that I noticed in a previous post (see the very end) that the exponential sum of Friedlander-Iwaniec, estimated by Birch and Bombieri, is (for prime moduli) just a special case of the general “correlation sums” that appeared in my recent work with Fouvry and Ph. Michel — in particular, our arguments (based on the sheaf-theoretic Fourier transform of Deligne, Laumon, Katz and others) give a conceptually simple proof of that estimate (I just wrote it down in a short separate note);

And although it doesn’t seem that Zhang uses it directly, I’d like to mention that the result of Friedlander and Iwaniec concerning the exponent of distribution of $latex d_3$ was improved by Heath-Brown a few years later, and that Fouvry, Michel and I very recently improved it quite a bit further (for prime moduli; the second part of that paper involves another application of the Bombieri-Friedlander-Iwaniec techniques to improve the exponent of distribution on average…)

And a philosophical preliminary conclusion, before diving into the work of Zhang: it is thrilling to see this result, and I particularly like that it comes completely unexpectedly, and yet uses all these beautiful ideas and methods from this analytic number theory that I love!

 

The Spring menagerie

I think readers can legitimately complain that not only have I not added a new post for a long time, but more schockingly, my last animal-related one goes back more than one year. So, to celebrate the recent belated aperçus of spring in Zürich and around, here are some pictures:

The first two are cheating, since they come from the Masoala Hall — but the first one illustrates the beautiful views from the very new canopy walk:

while the second is a rarely-seen lizard

Next comes a well-camouflaged bird, this one from a park in Graz

and another one from the aforementioned canopy

after which come a frog,

a snail,

and more frogs:

Hopefully more animal pictures will come before a year passes!

A missing word

From the blog of the rare books collection of the ETH Library, I just learnt that the word for the study and classification of grape species that I was looking for is “ampelography” (ampélographie in French).

(The relevance of this word to my daily life is that the computers on my home network are named after grapes; red grapes are reserved for desktops and white for laptops.)

Another exercise with characters

While thinking about something else, I noticed recently the following result, which is certainly not new:

Let $latex G$ be a compact topological group [ADDITIONAL ASSUMPTION pointed out by Y. Choi: connected, Lie group], and let $latex \rho$ be a finite-dimensional irreducible unitary continuous representation of $latex G$ on a vector space $latex V$. Then the natural representation $latex \pi$ of $latex G$ on $latex \mathrm{End}(V)$ decomposes as a direct sum of one-dimensional characters if and only if $latex \rho$ is of dimension $latex 1$.

One direction is clear: if $latex \rho$ has dimension one, then $latex \pi$ is simply the trivial one-dimensional representation. For the converse, here is an argument with character theory.

As a first step, note that if $latex \rho$ (of dimension $latex d\geq 1$, say) has this property, then in fact $latex \pi$ decomposes as a direct sum of distinct one-dimensional characters: indeed, the multiplicity of a character $latex \chi$ in $latex \pi$ is the same as
$latex n_{\chi}=\int_{G}\chi(x)\mathrm{Tr}(\pi(g))dg,$
where $latex dg$ is the probability Haar measure on $latex G$, and since
$latex \mathrm{Tr}(\pi(g))=|\mathrm{Tr}(\rho(g))|^2,$
we get
$latex n_{\chi}\leq \int_{G}\mathrm{Tr}(\pi(g))dg=1$
by the orthogonality relations of characters. (Algebraically, this is just an application of Schur’s lemma).

Thus if we decompose $latex \pi$ into irreducible representations, we get
$latex \pi=\bigoplus_{1\leq i\leq d^2} \chi_i,$
where the $latex \chi_i$ are distinct one-dimensional characters. We then know by orthogonality that
$latex d^2=\int_{G} |\mathrm{Tr}(\pi(g))|^2 dg=\int_{G} |\mathrm{Tr}(\rho(g))|^4 dg.$

Now the last-integral is bounded by
$latex \int_{G} |\mathrm{Tr}(\rho(g))|^4 dg\leq \mathrm{Max}_{g}|\mathrm{Tr}(\rho(g))|^2 \times \int_G|\mathrm{Tr}(\rho(g))|^2dg\leq d^2,$
(since $latex |\mathrm{Tr}(\rho(g))|\leq d$). Comparing, this means that there must be equality throughout in this estimate, which in turn implies that $latex |\mathrm{Tr}(\rho(g))|=d$ for all $latex g\in G$. Since $latex \rho(g)$ is unitary of size $latex d$, this implies that $latex \rho(g)$ is scalar for all $latex g$, and since it is assumed to be irreducible, it is in fact one-dimensional.

I see two interesting points in this argument: (1) is there a purely algebraic proof of the last part? I haven’t thought very hard about this yet, but it would be nice to have one; (2) the appearance of the fourth moment of $latex \rho$ is nicely reminiscent of the Larsen alternative (see Section 6.3 of my notes on representation theory, for instance…)