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Number Theory Seminar

Gaps distribution for 1D Fourier Quasi Crystals

Lior Alon - Princeton University

Location:  In person, Hill Center room 525 and Zoom
Date & time: Tuesday, 12 April 2022 at 2:00PM - 3:00PM

Abstract: 

The notion of a crystal, in physics, usually concerns a material whose atoms are ordered periodically. Mathematically, crystals are modeled by periodic discrete (i.e. atomic) measures. A key feature of a periodic discrete measure is that its Fourier transform is also a periodic atomic measure. This feature is important for crystallography since diffraction experiments measure the Fourier transform, from which they deduce the atomic structure of a given material. Roughly speaking, a quasicrystal is a material with a “quasi-periodic” atomic structure, in which case its Fourier transform is also discrete. Dan Shechtman won the 2011 Nobel prize for his discovery of the first quasicrystal. Mathematically, “quasi-periodic” periodic atomic measures were discussed prior to Shechtman’s discovery, and are of much interest, surprisingly, in one dimension rather than three dimensions, as Freeman Dyson suggested in 2009 (Birds and Frogs). A crystalline measure on the reals, is a discrete tempered measure whose Fourier transform is also discrete. The term “tempered” means that it cannot grow too fast on large intervals. A Fourier Quasi Crystal (in short FQC) is a crystalline measure with an additional growth estimate on the absolute value of its coefficients and the coefficients of its Fourier transform. We say that a measure is a unit mass FQC if all its coefficients equal 1. In 2020 Kurasov and Sarnak constructed unit mass FQC in terms of certain stable polynomials. Soon after, Olevskii and Ulanovskii showed that every unit mass FQC can be constructed in a similar manner. I will present a recent work in progress, on the characterization of these unit mass FQC in terms of the stable polynomials, applying tools from real algebraic geometry and ergodic theory. In particular, I will show that the gaps distribution of such a discrete measure can be calculated analytically in terms of the zero set of the associated polynomial, and that a repulsion phenomenon can be observed. That is, the proportion of small gaps goes to zero with the gap size. 

The talk will be in person and broadcast via zoom, please email for zoom information.

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