## 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…)
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.
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 $1/(\pi |h|)$.)

## 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.

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.

## Two biographies

Coincidentally, I finished reading two biographies in the last few days: R. Ellman’s “James Joyce” and R. D.G. Kelley’s “Thelonious Monk: The Life and Times of American Original”. Although I admire both Joyce and Monk, there is no question who is my favorite: on a desert island, I would take Monk’s records with me. And yet, I read through Joyce’s biography in barely more than a week, and only read Monk’s rather slowly over a few months — obviously, it is much easier to write about the life of a writer, quoting liberally from his letters and limericks, than to write about a musician without an accompanying CD or recording.

P.S. Ellman’s biography at least convincingly corrects the story I mentioned in an earlier post about Joyce moving frequently in Zürich because of his inability to pay the rent: his Zürich years during the first World War coincide with the time when he finally got a decent income (in principle) to not have to worry about such things. It seems however that he was very inventive in dealing with creditors earlier in Trieste…

P.P.S. The first name “Thelonious” apparently comes from Saint Tillo or Théau or Tillonius, who was active in the late 7th Century in Flanders and France; his feast day is January 7th, and he is noted for having cured the Bishop Hermenus of Limoges by informing him that he (Tillonius) was dying and requested him (the Bishop) to come and bury him.

## Définition tendancieuse

(I guess the title of this post would translate as something like “Biased definition” in English; according to the OED, “tendencious” does exist, but is ascribed as coming from the German “tendenziös”)

My son is currently reading an abridged version of Les Misérables for his French class. This is a text intended for schools and comes (among other things) with explanations of “hard words”. While glancing through it recently, I noticed the following striking instance:

Le hasard, c’est-à-dire la providence(1)

where the footnote translates, in lapidary style:

1. Providence = chance

(in English: Providence = luck). I may not know a lot about Victor Hugo, but it’s as clear as day to me that nothing could be further away from his use of the word “providence” than the idea that this is mere luck.

This reminded me of another definition I have seen in the French Larousse Universel encyclopedic dictionary from 1922 concerning the German language (see here in the middle of the page):

Langue: … une langue laborieuse… de là un certain manque de rapidité et de précision dans l’expression de la pensée.

(Or: … a clumsy language… from this comes a certain lack of speed and precision in the expression of one’s thoughts.)

This is actually a very nice book overall, with wonderfully useful illustrations to understand what, say, a “face-à-main” is, or to remind yourself of the important classification of “chapeaux bicornes” 