The support of Kloosterman paths

Will Sawin and I just put up on arXiv a preprint that is the natural follow-up to our paper on those most alluring of shapes, the Kloosterman paths.

As the title indicates, we are looking this time at the support of the limiting random Fourier series that arose in that first paper, namely
K(t)=t\mathrm{ST}_0+\sum_{h\not=0}\mathrm{ST}_h\frac{e^{2i\pi ht }-1}{2i\pi h},
where (\mathrm{ST}_h)_{h\in\mathbf{Z}} is a sequence of independent Sato-Tate-distributed random variables. In a strict sense, this should be a very short paper, since the computation of the support is easily achieved using some basic probability and elementary properties of Fourier series: it is the set of continuous functions f\colon [0,1]\to \mathbf{C} such that (1) the value of f at t=1 is real and belongs to [-2,2]; (2) the function g(t)=f(t)-tf(1) has purely imaginary Fourier coefficients \hat{g}(h) for h\not=0; (3) we have |\hat{g}(h)|\leq 1/(\pi |h|) for all h\not=0.

So why is the paper 26 pages long? The reason is that this support (call it \mathcal{S}) is a rather interesting set of functions, and we spend the rest of the paper exploring some of its properties. Most importantly, the support is not all functions, so we can play the game of picking our favorite continuous function on [0,1] (say f_0) and ask whether or not f_0 belongs to \mathcal{S}.

For instance:

  1. Fixing a prime p_0, and a_0, b_0 invertible modulo p_0, does the Kloosterman path K_{p_0}(a_0,b_0) itself belong to the support? Simple computations show that it depends on (p_0,a_0,b_0)! For instance, the path for the Kloosterman sum \mathrm{Kl}_2(8,1;9), shown below, does not belong to the support. (As we observe, it looks like a Shadok, whose mathematical abilities are well-known — sorry, the last link is only in French ; I suggest to every French-aware reader to watch the corresponding episode, since the voice of C. Piéplu achieves the seemingly impossible in making this hilarious text even funnier…)
    Kl_2(8,1;19)
    Kl_2(8,1;19)
  2. On the other hand, the path giving the graph of the Takagi function T (namely f(t)=t+iT(t)) belongs to the support.
    Takagi function
    Takagi function
  3. But maybe the most interesting problem from a mathematical point of view is one of pure analysis: when we see a Kloosterman path (such as the one above), we only see its image as a function from [0,1] to \mathbf{C}, independently of the parameterization of the path. So we can take any shape in the plane that can be represented as the image of a function f satisfying the conditions (1) and (2) above, and ask: is there a reparameterization of f that belongs to the suppport? For instance, for the Kloosterman paths themselves (as in (1) above), it is not difficult to find one: instead of following each of the p_0-1 segments making the Kloosterman path in time 1/(p_0-1), one can insert a “pause” of length 1/(2p_0) at the beginning and end of the path, and then divide equally the remaining time for the p_0-1 segments. (The fact that this re-parameterized path, whose image is still the same Kloosterman path, belongs to the support \mathcal{S} is then an elementary consequence of the Weil bound for Kloosterman sums).

  4. In general, the question is whether a given f has a reparameterization with Fourier coefficients (rather, those of t\mapsto f(t)-tf(1)) are all smaller than 1/(\pi |h|). This is an intriguing problem, and looking into it brought us into contact with some very nice classical questions in Fourier analysis, that I discuss in this later post. We only succeeded in proving the existence of a suitable reparameterization for real-valued functions, for reasons explained in the aforementioned later post, and it is an interesting analysis problem whether the result holds for all functions. A positive answer would in particular settle another natural question that we haven’t been able to handle yet: is there a space-filling curve in the support of the Kloosterman paths?

All this is great analytic fun. But there are nice arithmetic consequences of our result. By the definition of the support, we know at least that any f\in\mathcal{S} has the property that, with positive probability, the actual path of the partial sums of the Kloosterman sums will come as close as we want (uniformly on [0,1]) to f, and this is an arithmetic statement. For instance, simply because the zero function belongs to the support, we deduce that, for a large prime p, there is a positive proportion of (a,b)\in \mathbf{F}_p^{\times}\times\mathbf{F}_p^{\times} such that all partial sums
\frac{1}{\sqrt{p}}\sum_{1\leq x\leq j}\exp(2i\pi(ax+b\bar{x})/p),
for 1\leq j\leq p-1, have modulus <\varepsilon.
In other words, there is a non-zero probability that all the normalized partial sums of the Kloosterman sums are very small. (It is interesting to note that this is emphetically not true for character sums… the point is that their Fourier expansion involves multiplicative coefficients, so they cannot become smaller than 1/(\pi |h|).)

Tornare a Ventotene

I participated last week to the wonderful Ventotene 2017 Conference, a worthy continuation of Ventotene 2015. Reaching the island required this time even more of the stamina that the conference website recommends, since the weather was rough enough that the faster hydrofoil boat did not run (stranding about 30 of the participants in Formia on Sunday evening), while even the rather bigger one behaved more like a large scale roller-coaster than most people would wish.

After arriving at the island, Monday was still a bit unpleasant (it was much more for those who were unlucky to be exposed when one of the few short but very violent rain showers fell…), but the remaining of the time was beautiful. On the way back, I had to stop in Rome for a night, and tasted the most delicious ragù bianco di coniglio that one can imagine.

Bird
Bird
Sun
Sun
Cat
Cat
Lizard
Lizard
Festive balloon
Festive balloon

I’m already looking forward to the next conference…

A dragon of genus one

I just came back from Hong Kong, where I participated in a conference on automorphic forms and all their applications. Among the talks, I especially enjoyed to hearing about the current status of the Jacquet-Langlands correspondance from Badulescu, and also the definition and properties of uniform pro-p-groups in the talk of Jiu-Kang Yu.

I stayed two days after the conference and had time to do some visiting and shopping. My favorite browsing time was in the Yue Hwa Chinese Emporium (the address was suggested by Yuk-Kam Lau, who also suggested the restaurant where we had our best dinner). I couldn’t resist buying the torus-shaped box that I saw there.

Dragon torus box
Dragon torus box

According to my wife, this is certainly intended to hold necklaces and bracelets. It is decorated with a dragon and what I think is a phoenix (maybe representing two elements of a basis of the homology or cohomology of the torus). I haven’t yet measured it to check if it has complex multiplication.

I took a few animal pictures, although the rather oppressive weather made it a bit difficult to spend time walking leisurely outside.

Birds
Birds
Raptor
Raptor
Dragonfly
Dragonfly

And I saw a rather melancholy piano in a side street…

Piano
Piano

Musically, I think there is a minor modernist masterpiece to be written based on the rings, whistles and noises of the Octopus card system, recorded at one of the more important exits of the MTR subway at some of its busier hours of operation.

Who wrote the “New Oxford Shakespeare”?

At the very least, nobody from the University of Oxford (except if some of the Anonymous collaborators of some of the plays were professors there). Indeed, all the editors listed on the covers come from other institutions.

In comparison with the 1987 edition (called more modestly, if apparently inaccurately, “The complete works”), the new version identifies more works where Shakespeare was involved, and (taking from the other hand) finds also more plays where other writers participated. This is all explained in fair detail in a companion book full of statistical studies of proportions of rhymes or of feminine endings, or other fine points of prosody. Maybe most interesting (to me) is the play “Arden of Fevershame” that is now attributed in part to Shakespeare at the very beginning of his career, since its theme (the story of a then fairly recent murder most foul committed among England commoners) is rather far from the themes of most of his other plays.

The impressive volumes also make for excellent book-ends.

Books aligned
Books aligned

And there is still apparently a further volume (or two) to come, of “alternative versions” of those plays that are known in two or more substantially different early texts (e.g., “King Lear”).

I am eagerly anticipating a similarly ambitious scholarly N.O.W (New Oxford Wodehouse); in fact, I am happy to volunteer for the exacting role of editor of the Jeeves & Wooster canon. Or, if objections are raised against the attribution of such a crucial part of the oeuvre to a Frenchman of Polish and Breton origins, I will gladly take responsibility for the volumes encompassing the acts of the fifth Earl of Ickenham, fewer in number but by no means in importance.