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.).
]]>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…
]]>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
where 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 such that (1) the value of at is real and belongs to ; (2) the function has purely imaginary Fourier coefficients for ; (3) we have for all .
So why is the paper 26 pages long? The reason is that this support (call it ) 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 (say ) and ask whether or not belongs to .
For instance:
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 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 ) to , and this is an arithmetic statement. For instance, simply because the zero function belongs to the support, we deduce that, for a large prime , there is a positive proportion of such that all partial sums
for , have modulus .
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 .)
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.
I’m already looking forward to the next conference…
]]>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.
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.
And I saw a rather melancholy piano in a side street…
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.
]]>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.
]]>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.
]]>The Norwegian Academy of Science and Letters announces the 2017 Abel Prize is awarded to Yves Meyer, École normale supérieure Paris-Saclay, France…
(from the AMS home page, as of right now…)
I’m even prouder now than before of having bought Meyer’s “Wavelets and operators” in the Peoria, Illinois, science museum store…
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