Lectures given at the Workshop on Fourier Analysis and Convexity, Università Di Milano–bicocca

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Let E be a measurable set in Rn such that 0 < |E | < ∞. We will say that E tiles Rn by translations if there is a discrete set T ⊂ Rn such that, up to sets of measure 0, the sets E + t: t ∈ T are mutually disjoint and ⋃

This article is based on the series of lectures on the interaction of Fourier analysis and geometric combinatorics delivered by the author in Padova at the Minicorsi di Analisi Matematica in June, 2002.

Abstract. We introduce a duality for Affine Iterated Function Systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure µ, and we ask when µ is a spectral measure, i.e., when the Hilbert space L2 (µ) has an orthonormal basis (ONB) of exponentials {eλ | λ ∈ Λ}. We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in

Spectra of certain types of polynomials and tiling of integers with translates of finite sets

We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations. Mathematics Subject Classification: 42A99. 1 The results A Borel set Ω ⊂ Rn of positive measure is said to tile Rn by translations if there is a discrete set T ⊂ Rn such that, up to sets of measure 0, the sets Ω+t, t ∈ T, are disjoint and ⋃ t∈T (Ω+t) = R n. We may rescale Ω so that |Ω | = 1. We say that Λ = {λk: k ∈ Z} ⊂ R n is a spectrum for Ω if: {e 2πiλk·x}k∈Z is an orthonormal basis for L 2 (Ω). (1.1) A spectral set is a domain Ω ∈ Rn such that (1.1) holds for some Λ. Fuglede [2] conjectured that a domain Ω ⊂ Rn is a spectral set if and only if it tiles Rn by translations, and proved this conjecture under the assumption that either Λ or T is a lattice. The conjecture is related to the question of the existence of commuting self-adjoint extensions of the operators −i ∂

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The spectral set conjecture and multiplicative properties of roots of polynomials

The spectral set conjecture and multiplicative properties of roots of polynomials