Introduction to approximate groups

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Hyperuniformity and non-hyperuniformity of quasicrystals

We develop a general framework to study hyperuniformity of various mathematical models of quasicrystals. Using this framework we provide examples of non-hyperuniform quasicrystals which unlike previous examples are not limit-quasiperiodic. Some of these examples are even anti-hyperuniform or have a positive asymptotic number variance. On the other hand we establish hyperuniformity for a large class of mathematical quasicrystals in Euclidean spaces of arbitrary dimension. For certain models of quasicrystals we moreover establish that hyperuniformity holds for a generic choice of the underlying parameters. For quasicrystals arising from the cut-and-project method we conclude that their hyperuniformity depends on subtle diophantine properties of the underlying lattice and window and is by no means automatic

Spectral theory of approximate lattices in nilpotent Lie groups

We consider approximate lattices in nilpotent Lie groups. With every such approximate lattice one can associate a hull dynamical system and, to every invariant measure of this system, a corresponding unitary representation. Our results concern both the spectral theory of the representation and the topological dynamics of the system. On the spectral side we construct explicit eigenfunctions for a large collection of central characters using weighted periodization against a twisted fiber density function. We construct this density function by establishing a parametric version of the Bombieri-Taylor conjecture and apply our results to locate high-intensity Bragg peaks in the central diffraction of an approximate lattice. On the topological side we show that under some mild regularity conditions the hull of an approximate lattice admits a sequence of continuous horizontal factors, where the final horizontal factor is abelian and each intermediate factor corresponds to a central extension. We apply this to extend theorems of Meyer and Dani-Navada concerning number-theoretic properties of Meyer sets to the nilpotent setting

Approximate Invariance for Ergodic Actions of Amenable Groups

We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups. As an application of these techniques, we prove a dynamical generalization of Kneser\u27s celebrated density theorem for subsets in (Z, +), valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study

Expressions for forces and torques in molecular simulations using rigid bodies

Expressions for intermolecular forces and torques, derived from pair potentials between rigid non-spherical units, are presented. The aim is to give compact and clear expressions, which are easily generalised, and which minimise the risk of error in writing molecular dynamics simulation programs. It is anticipated that these expressions will be useful in the simulation of liquid crystalline systems, and in coarse-grained modelling of macromolecules

Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula

We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions. (C) 2021 The Author(s). Published by Elsevier Inc

Sets of transfer times with small densities

In this paper we introduce and discuss various notions of doubling for measurepreserving actions of a countable abelian group G. Our main result characterizes 2-doubling actions, and can be viewed as an ergodic-theoretical extension of some classical density theorems for sumsets by Kneser. All of our results are completely sharp and they are new already in the case when G = (Z; +)

Infection avoidance behavior: Viral exposure reduces the motivation to forage in female Drosophila melanogaster

Infection avoidance behaviors are the first line of defense against pathogenic encounters. Behavioral plasticity in response to internal or external cues of infection can therefore generate potentially significant heterogeneity in infection. We tested whether Drosophila melanogaster exhibits infection avoidance behavior, and whether this behavior is modified by prior exposure to Drosophila C Virus (DCV) and by the risk of DCV encounter. We examined 2 measures of infection avoidance: (1) the motivation to seek out food sources in the presence of an infection risk and (2) the preference to land on a clean food source over a potentially infectious source. While we found no evidence for preference of clean food sources over potentially infectious ones, previously exposed female flies showed lower motivation to pick a food source when presented with a risk of encountering DCV. We discuss the relevance of behavioral plasticity during foraging for host fitness and pathogen spread

The Evolution of Economic Governance in EMU

This paper examines the benefits of co-ordination in EMU in a stylised manner and how these benefits have shaped the co-ordination framework in EMU. It then discusses in detail the co-ordination experience in four areas that are particularly important for the functioning of EMU: (i) fiscal policy co-ordination under the Stability and Growth Pact (SGP); (ii) the co-ordination of structural policies under the Lisbon Strategy for Growth and Jobs; (iii) the representation and co-ordination of euro-area positions in international financial fora; and (iv) the co-ordination of macroeconomic statistics. The thrust of the findings is that EMU's system of economic governance has, overall, proven fit for purpose. The current policy assignment to the institutions and instruments that govern the conduct of economic policy in EMU is sound, even though further progress is necessary in several areas, particularly as regards external representation.Governance, EMU, euro area, co-ordination, van den Noord, D�hring, Langedijk, Nogueira-Martins,Pench, Temprano-Arroyo, Thiel

Central limit theorems for Diophantine approximants

\ua9 2019, The Author(s). In this paper we study certain counting functions which represent the numbers of solutions of systems of linear inequalities arising in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior that these functions exhibit and prove a central limit theorem for them. Our approach is based on a quantitative study of higher-order correlations for functions defined on the space of lattices and a novel technique for estimating cumulants of Siegel transforms