When, exactly two years ago, I published my earlier post containing the story of J. Ménard, I was apparently suspected by some people of being the author of that text. I tried for a long time to find the original Spanish version mentioned in the text, whose existence conclusively refutes this assertion (since my understanding of Spanish is, unfortunately, non-existent). After much effort, I have finally succeeded!
This afternoon, while chatting with Will Sawin, between addressing rather technical points of ongoing projects, we observed that although we’ve seen seminar talks shared between two speakers, it was never with simultaneous speakers. It was just a step to jump from there to the idea that someone should write an Opera about a mathematical seminar talk, which — as opera does — would allow a duet, or trio, or quartet, or quintet, or sextet, of simultaneous speakers, including maybe some from the audience, or the chairperson trying to control the situation.
Unfortunately, I don’t know music, but if I were twenty years old, I’d be very tempted to write “Seminar”, the definitive opera about a math talk. At least, I can think of a libretto and try to write it (which language? I think French is best here, although the title should then be “Séminaire” or “L’exposé” in that case; which topic? good question — of course it would have to be a real talk; which style?)
In any case, I can safely predict a triumph in enlightened circles.
I already discussed the very interesting mathematics related to the Bohr-Pál Theorem that I learnt about while writing my preprint with W. Sawin on the support of Kloosterman path. This also led us (very) tangentially to what turns out to be an open geometric problem: Mihalik and Wieczorek asked whether there exists a continuous function which sends every interval in to a convex subset of , and whose image is not contained in an affine line in . In other words: suppose is continuous and satisfies the intermediate value property (in the sense that for any , any point on the segment between and is of the form for some between and ); is the image of contained in some affine line?
The existence of such functions may seem unlikely, but experience shows that being unlikely to exist has rarely stopped functions from actually existing.
The arithmetic relevance of a hypothetical Mihalik-Wieczorek function is that, by adapting Sahakyan’s argument (as discussed before), it seems that we would be able to construct a function in the support of Kloosterman paths that has image containing an open set, answering one of our lingering questions.
However, the question remains apparently open. The best known result (that we know about) is due to Pach and Rogers (1982), and independently Vince and Wilson (1984): there exists , with image not contained in an affine line, such that and are convex for all . (Note that Vince and Wilson conjecture at the end of their paper that a Mihalik-Wieczorek curve does not exist; however their argument is based on a weaker conjecture that this existence would imply, and that statement in turn is actually true, as was explicitly stated by Pach and Rogers at the end of their article; see this American Math. Monthly problem.).
Update (May 13, 2018) I’ve written a short note that checks in detail the statement above: if a Mihalik-Wieczorek curve exists, then there exists a space-filling curve in the support of the Kloosterman paths. (More precisely, this is done without the symmetry requirement of the Kloosterman paths, but these can be ensured by scaling the given curve to and then reflecting it on ). This note can also be seen as a detailed account of Sahakian’s Theorem…
Keen-memoried readers will remember the word appearing before on this blog. As one of the happy few who have read “The Unconsoled” twice, I applaud with pleasure the honor given to K. Ishiguro! (If my credentials are disputed, let me clearly state that I can answer the question: “Which spectacular goal scored by a Dutch player during the 1978 World Cup is described in the book?” — or rather, almost, since the description is ambiguous and could apply to two goals by the same player, during different games; I actually remember watching at least one of them).
In the course of writing our paper on the support of the Kloosterman paths, Will Sawin and I encountered some very beautiful classical questions of Fourier analysis. As discussed in the previous post, we were interested in the following question: given a continuous function , such that (1) we have and (2) the function has purely imaginary Fourier coefficients for , does there exist an increasing homeomorphism , such that , and such that the new function , in addition to the analogue of (1) and (2), also satisfies for ?
After some searching we found the right keywords, or mathematical attractor, for this type of problem. It starts with a short paper of Gyula, alias Julius, alias Jules, Pál, alias Pal, alias Perl, in 1914, who attributes to Fejér the following question : can one reparameterize a continuous periodic function in such a way that the new function has uniformly convergent Fourier series? Pál shows that he can almost do it: his reparameterized function has Fourier series that converges uniformly on , where can be arbitrarily small (but the change of variable depends on ).
This Theorem of Pál becomes the Bohr–Pál Theorem after Harald Bohr’s full answer to Fejér’s question in Acta Universitatis Szegediensis in 1935. (Note: I had never looked before at the archive of this particular journal; like most mathematical journals of this period, it is amazing how many names, and even theorems, one can recognize in almost any issue !)
Then comes a beautiful very short proof by Salem in 1944 (it is also the proof found in Zygmund’s book on trigonometric series). All three proofs rely on two results to achieve uniform convergence: one is Riemann’s mapping theorem, and the second is a result of Fejér, according to which the conformal map from the unit disc to the “interior” of a simple closed curve has the property that the boundary Fourier series converges uniformly. In fact, one can certainly write beautiful exercises or exams based on Salem’s proof, for a course in complex analysis that goes as far as Riemann’s Theorem…
The story does not stop here. It is amusing to note that none of Pál, Bohr or Salem feel that it is necessary to point out that they work with real-valued periodic functions. But this is essential to the basic structure of their argument (which starts by viewing as the imaginary part of a complex number that describes a simple closed curve in , the trick being to define the real part in a suitable way). Whether the Bohr-Pál Theorem holds for complex-valued functions was an open problem until the late 1970’s (see for instance p. 128 of this 1974 survey of Zalcman about real and complex problems in analysis), when it was proved by Kahane and Katznelson with a completely different approach. They in fact succeeded in proving that one can find a single change of variable that transforms simultaneously all functions in a compact set of the space of (real-valued periodic) continuous functions into functions with uniformly convergent Fourier series; applied to the real and imaginary parts of a complex-valued function, this gives the required extension.
Coming back to our original question, the complex-variable proofs give some information on the Fourier coefficients of the reparameterized function , precisely they imply that
which is encouraging, even if no pointwise estimate follows. But some variants of the question have been studied that get much closer to what we want. They are discussed (among other things) in a nice survey by Olevskii. I find especially fascinating one theorem of Olevskii himself, published in 1981: it is not always possible to find a change of variable (of a real-valued periodic continuous function) so that the reparameterized function has an absolutely convergent Fourier series. This answers a question of Luzin, and is explained in Section 3 of Olevskii’s survey. Interestingly, this problem is now, of course, easier for complex-valued functions, compared to real-valued functions, and it was also solved by Kahane and Katznelson in the -valued case.
The result that turned out to be most useful for our purposes is
due to Sahakyan (whose name is also spelled Saakjan or Saakyan). He proves the existence of reparameterizations of a continuous periodic function satisfying various asymptotic bounds for their Fourier coefficients. The result is here also restricted to real-valued functions, and again for a clear reason: the construction relies crucially on the intermediate value theorem for continuous functions. More precisely, Sahakyan’s idea is to use the Faber-Schauder expansions of continuous functions, and to find a change of variable that leads to a function with a “sparse” Faber-Schauder expansion, from which he estimates directly the Fourier coefficients using bounds for those of the Faber-Schauder functions. (I actually didn’t know about Faber-Schauder expansions before; I will also certainly make use of them for analysis or functional analysis exercises later…) This works because the Faber-Schauder coefficients are quite simple: they are of the form
for suitable real numbers . One can now easily imagine how the intermediate value theorem will make it possible to find a change of variable to make such a coefficient vanish.
Using Sahakyan’s main lemma, with a few modifications to take into account the additional symmetry we require, we were able to solve our reparameterization problem for the support of Kloosterman paths, in the case of real-valued functions. The complex-value case, as far as we know, remains open…