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Register a new userPure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity
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We show that a translation bounded measure has pure point diffraction if and only if it is mean almost periodic. We then go on and show that a translation bounded measure solves what we call the phase problem if and only if it is Besicovitch almost periodic. Finally, we show that a translation bounded measure solves the phase problem independent of the underlying van Hove sequence if and only if it is Weyl almost periodic. These results solve fundamental issues in the theory of pure point diffraction and answer questions of Lagarias.
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We consider metrizable ergodic topological dynamical systems over locally compact, $sigma$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order.
Fourier-transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier--Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.
We briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-and-project sets, which is based on Bochners theorem from Fourier analysis. This clarifies a common view that the diffraction of a quasicrystal is determined by the diffraction of its underlying lattice. To illustrate our approach, we will also treat a number of well-known explicitly solvable examples.
The matrix $A:mathbb{R}^n to mathbb{R}^m$ is $(delta,k)$-regular if for any $k$-sparse vector $x$, $$ left| |Ax|_2^2-|x|_2^2right| leq delta sqrt{k} |x|_2^2. $$ We show that if $A$ is $(delta,k)$-regular for $1 leq k leq 1/delta^2$, then by multiplying the columns of $A$ by independent random signs, the resulting random ensemble $A_epsilon$ acts on an arbitrary subset $T subset mathbb{R}^n$ (almost) as if it were gaussian, and with the optimal probability estimate: if $ell_*(T)$ is the gaussian mean-width of $T$ and $d_T=sup_{t in T} |t|_2$, then with probability at least $1-2exp(-c(ell_*(T)/d_T)^2)$, $$ sup_{t in T} left| |A_epsilon t|_2^2-|t|_2^2 right| leq Cleft(Lambda d_T deltaell_*(T)+(delta ell_*(T))^2 right), $$ where $Lambda=max{1,delta^2log(ndelta^2)}$. This estimate is optimal for $0<delta leq 1/sqrt{log n}$.
The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibres. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibres have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.
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