Archive for March, 2016

Poisson, Guinand, Meyer

March 27, 2016

[Posted on march 27, 2016]

A recent highly interesting paper by Yves Meyer (PNAS paywalled, local version at ENS Cachan, and seminar notes) constructs explicitly new Poisson-type summation formulas (building on previous little known work of Andrew-Paul Guinand  and an existence result of Nir Lev and Aleksander Olevskii) : the big difference with Poisson summation is that the new formulas do not have support on a lattice but only on a locally finite set (and then provide new examples of crystalline measures).

Since these new results involve some arithmetic (see below) I’ve asked over at MO whether this was known to number theorists, but there hasn’t been any immediate answer, so perhaps not and there’s probably room for interesting further work on the topic.

To state things very explicitely (for my own benefit, but also just for the beauty of it), here are the formulas taken directly from Meyer’s paper :

Poisson (Dirac comb case): on a lattice \Gamma\subset\mathbb{R}^n and its dual \Gamma^* we have for any function f in the Schwartz class \mathcal{S}(\mathbb{R}^n) that

\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma}f(\gamma) = \sum_{\eta \in\Gamma^*}\widehat{f}(\eta)

Poisson (corollary of Dirac comb case) : for every \alpha,\beta \in\mathbb{R}^n we have (in terms of distributions to make the comb more explicit still)

\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma +\alpha} e^{2i\pi\beta .\gamma}\delta_{\gamma} = e^{2i\pi\alpha .\beta} \sum_{\eta \in\Gamma^* +\beta} e^{2i\pi\alpha .\eta}\delta_{\eta}

Guinand : define for any n\in\mathbb{N} the number of sums of three squares that equal to n by r_3(n) (by Legendre’s theorem this is possible only for those n not of the form 4^j(8k+7)). Then introducing Guinand’s distribution (acting on functions of the variable t)

\displaystyle \sigma := -2\frac{d}{dt}\delta_0 + \sum_{n=1}^{+\infty} \frac{r_3(n)}{\sqrt{n}} (\delta_{\sqrt{n}}-\delta_{-\sqrt{n}})

then we have \langle \sigma ,f\rangle = \langle -i\sigma ,\widehat{f}\rangle .

Meyer (first example) : introducing the function \chi on \mathbb{N} (this is a clash of notation with Dirichlet characters) by \chi(n)=-\frac{1}{2} when n\not\equiv 0\pmod{4}, \chi (n)=4 when n\equiv 4\pmod{16} and \chi (n)= 0 when n\equiv 0\pmod{16} then with the distribution

\displaystyle \tau := \sum_{n=1}^{+\infty} \frac{\chi(n)r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}}-\delta_{-\frac{\sqrt{n}}{2}})

we have \langle \tau ,f\rangle = \langle -i\tau ,\widehat{f}\rangle .

The support of \sigma and \tau are thus defined as subsets of \{\pm\sqrt{n}|n\in\mathbb{N}\} and \{\pm\frac{\sqrt{n}}{2}|n\in\mathbb{N}\} by their respective arithmetic conditions, and thus are definitely not equally spaced lattice points.

Meyer (second example) : with the distribution

\displaystyle \rho := 2\pi\delta_{\frac{1}{2}} +2\pi\delta_{-\frac{1}{2}} + \sum_{n=1}^{+\infty} \frac{\sin(\pi\sqrt{n})r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{\frac{\sqrt{n}}{2}-\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}-\frac{1}{2}} )

we have \langle \rho ,f\rangle = \langle \rho ,\widehat{f}\rangle (very nice!).

There are several other examples in Meyer’s paper, as well as higher-dimensional constructions (that I haven’t absorbed yet, so I’ll stop here).

Update (march 27): two relevant papers I’ve just found

  • On the Number of Primitive Representations of Integers as Sums of Squares by Shaun Cooper and Michael Hirschhorn published in Ramanujan J (2007) 13:7–25, which in particular provides the explicit formula r_3(n)=\sum_{d^2|n}r_3^p\big ( \frac{n}{d^2}\big ) where the function r_3^p is in turn explicited (p is a label standing for ‘primitive’)
  • Irregular Poisson Type Summation by Yu. Lyabarskii and W.R. Madych, published in SAMPLING THEORY IN SIGNAL AND IMAGE PROCESSING Vol. 7, No. 2, May 2008, pp. 173-186, which does prove a Poisson-type formula with irregularly spaced sampling points (but if I understand well the examples they mention at the end show it is still different from the results of Guinand and Meyer, to be confirmed)
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Sphere packings and other news

March 23, 2016

[Posted on 23 march, 2016]

Apparently there have been recently two great advances on sphere packings in dimension higher than 3 :

  • a paper by Maryna Viasovska establishing the densest packing in dimension 8
  • another paper by Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska solving the 24-dimensional case (with the Leech lattice)

In other news :

Oranges, by dncnH on flickr

Early march items

March 5, 2016

[Posted on march 5, 2016]

For immediate consumption :

  • the latest issue of the Newsletter of the EMS is out, with several outstanding introductory pieces, a great read.   There’s also an article by Fabian Müller and Olaf Teschke who have measured the amount of published papers that are on the arXiv thanks to new tools provided by zbMATH. Among other things, they mention that  “for the publication year 2014, about 55% of research in algebraic geometry, algebraic topology and K-Theory is available through the arXiv but only about 10% in numerical mathematics and 1% in mathematics history or mathematics education“. I’m slightly surprised that the algebraic geometry figure isn’t into the 70-80% territory.
  • as most people know by now, Discrete Analysis, the arXiv orverlay journal managed by Sir Timothy Gowers aimed at challenging for-profit publishers, is now online (and has lots of nice features)
  • CNRS medals in mathematics have been awarded : a Silver Medal to Isabelle Gallagher (PDEs, in particular Navier-Stokes), and a Bronze Medal to Simon Riche (Geometric Representation Theory)
  • the 2015-2016 Cours Peccot will be given by Nicolas Curien (next may at Collège de France) and by Marco Robalo (TBA)
  • the 2016 Abel Prize is due to be announced soon, on march 15
  • Olivia Caramello has chosen to update her public controversy with category theorists
  • a 177 pages paper by Tye Lidman and Ciprian Manolescu identifies two Seiberg-Witten Floer homology theories. To the outsider, this looks very much like the length and depth of papers that cement the status of a future Fields Medalist. (As an anecdote, one can notice a math.SE question in the bibliography — yes, math.SE not MO).
  • a word of caution taken from the website of the IHP conference in honor of Margulis’ 70th birthday next june, which applies to all other conferences in France in that period : “Due to the European Football Championship (EURO 2016) to be held in France, June 10th-July 10th 2016, participants are encouraged to book their hotel room a long time in advance“.

 

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Centre Pompidou Metz and the moon, winter 2016. Public domain.