E. Kowalski's blog

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A parity lemma of A. Irving

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In his recent work on the divisor function in arithmetic progressions to smooth moduli, A. Irving proves the following rather amusing lemma (see Lemma 4.5 in his paper):

Lemma Let p be an odd prime number, let k\geq 1 be an integer and let h=(h_1,\ldots,h_k) be a k-tuple of elements of \mathbf{F}_p. For any subset I of \{1,\ldots, k\}, denote
h_I=\sum_{i\in I}{h_i},
and for any x\in\mathbf{F}_p, let
\nu_h(x)=|\{I\subset \{1,\ldots, k\}\,\mid\, h_I=x\}|
denote the multiplicity of x among the (h_I).
Then if none of the h_i is zero, there exists some x for which \nu_h(x) is odd.

I will explain two proofs of this result, first Irving’s, and then one that I came up with. I’m tempted to guess that there is also a proof using some graph theory, but I didn’t succeed in crafting one yet.

Irving’s proof. This is very elegant. Let \xi be a primitive p-th root of unity. We proceed by contraposition, hence assume that all multiplicities \nu_h(x) are even. Now consider the element
of the cyclotomic field K_p=\mathbf{Q}(\xi). By expanding and using the assumption we see that
N=\sum_{x\in\mathbf{F}_p} \nu_h(x)\xi^{x}\in 2\mathbf{Z}[\xi].
In particular, the norm (from K_p to \mathbf{Q}) of N is an even integer, but because p is odd, the norm of 1+\xi^{h_i} is known to be odd for all h_i\not=0. Hence some factor must have h_i=0, as desired.

A second proof. When I heard of Irving’s Lemma, I didn’t have his paper at hand (or internet), so I tried to come up with a proof. Here’s the one I found, which is a bit longer but maybe easier to find by trial and error.

First we note that
\sum_{x\in \mathbf{F}_p} \nu_h(x)=2^k
is even. In particular, since p is odd, there is at least some x with \nu_h(x) even.

Now we argue by induction on k\geq 1. For k=1, the result is immediate: there are two potential sums 0 and h_1, and so if h_1\not=0, there is some odd multiplicity.

Now assume that k\geq 2 and that the result holds for all (k-1)-tuples. Let h be a k-tuple, with no h_i equal to zero, and which has all multiplicities \nu_h(x) even. We wish to derive a contradiction. For this, let j=(h_1,\ldots,h_{k-1}). For any x\in\mathbf{F}_p, we have
by counting separately those I with sum x which contain k or not.

Now take x such that \nu_j(x) is odd, which exists by induction. Our assumptions imply that \nu_j(x-h_k) is also odd. Then, iterating, we deduce that \nu_j(x-nh_k) is odd for all integers n\geq 0. But the map n\mapsto x-nh_k is surjective onto \mathbf{F}_p, since h_k is non-zero. Hence our assumption would imply that all multiplicities \nu_j(y) are odd, which we have seen is not the case… Hence we have a contradiction.

Written by Kowalski

March 16th, 2015 at 12:31 pm

Posted in Exercise,Mathematics

Who proved the Peter-Weyl theorem for compact groups?


Tamas Hausel just asked me (because of my previous post on the paper of Peter and Weyl) how could Peter and Weyl have proved the “Peter-Weyl Theorem” for compact groups in 1926, not having Haar measure at their disposal? Indeed, Haar’s work is from 1933! The answer is easy to find, although I had completely overlooked the point when reading the paper: Peter and Weyl assume that their compact group is a compact Lie group, which allows them to discuss Haar measure using differential forms!



So the question is: who first proved the full “Peter-Weyl” Theorem for all compact groups? Pontryaguin, in 1936, certainly does, without remarking that Peter-Weyl didn’t, possibly because it was clear to anyone that the argument would work as soon as an invariant measure was known to exist. But since there are “easier” proofs of the existence of Haar measure for compact groups than the general one for all locally-compact groups (using some kind of fixed-point argument), it is not inconceivable that someone (e.g., von Neumann) might have made the connection before.

In fact, there is an amusing mystery in connection with Pontryaguin’s paper and von Neumann: concerning Haar measure, he refers to a paper of von Neumann entitled Zum Haarschen Mass in topologischen Gruppen, and gives the helpful reference Compositio Math., Vol I, 1934. So we should be able to read this paper on Numdam? But no! The first volume of Compositio Mathematica there is from 1935; it is identified as Volume I, and there is no paper of von Neumann to be found…

[Update: as many people pointed out, the paper of von Neumann is indeed on Numdam, but appeared in 1935; I was tricked by the absence of 1934 on the Compositio archive and the author’s name being written J.V. Neumann (I had searched Numdam with “von Neumann” as author…)]

Written by Kowalski

March 15th, 2015 at 8:03 pm

Posted in Mathematics


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For those readers who understand spoken French (or simply appreciate the musicality of the language) and are interested in the history of mathematics, I warmly recommend listening to the recording of a recent programme of Radio France Internationale entitled “Pourquoi Bourbaki ?” In addition to the dialogue of Sophie Joubert with Michèle Audin and Antoine Chambert-Loir, one can hear some extracts of older émissions with L. Schwartz, A. Weil, H. Cartan, J. Dieudonné, for instance.

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February 27th, 2015 at 7:31 pm

Аналитическая теория чисел


Thanks to the recent Russian translation of my book with Henryk Iwaniec, I can now at least read my own last name in Cyrillic; I wonder what the two extra letters really mean…

Analytic Number Theory in Russian

Analytic Number Theory in Russian

Written by Kowalski

February 9th, 2015 at 9:06 pm

Posted in Language,Mathematics

More conferences

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It seems that most of my posts these days are devoted to announcing conferences in which I am involved as organizer… Indeed, there are two coming up this year (actually three, if I count the MSRI summer school):

(1) May 14 and 15, we will have the Number Theory Days 2015 at EPF Lausanne; the speakers are Gaetan Chenevier, Henryk Iwaniec, Alena Pirutka, Chris Skinner and Zhiwei Yun; this is co-organized by Ph. Michel and myself.

(2) Immediately afterward, from May 18 to 22, comes a conference at FIM, co-organized by H. Iwaniec, Ph. Michel and myself, with the title of “Analytic Aspects of Number Theory”; the current list of speakers is to be found on the web page; here is the poster (which is based on a picture taken by Henryk around Zürich last Fall):

AANT Poster

Analytic Aspects of Number Theory

Most importantly, there is a certain amount of funding available for local expenses of your researchers (doctoral and postdoctoral students). Applications can be made here (before Feburuary 6; the form states January 28, but this is an error that will be corrected).

Written by Kowalski

January 29th, 2015 at 3:56 pm

Posted in ETH,Mathematics