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Generalized Model Sets and Dynamical Systems
 CRM Monograph Series
, 1999
"... It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigmacompact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure ..."
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Cited by 64 (0 self)
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It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigmacompact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure point diffraction spectra.
Tiling the Line with Translates of One Tile
"... This paper shows for a bounded tile that all tilings it gives of R are periodic, and that there are finitely many translationequivalence classes of such tilings. The main result of the paper is that for any tiling of R by a bounded tile, any two tiles in the tiling differ by a rational multiple of ..."
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Cited by 46 (8 self)
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This paper shows for a bounded tile that all tilings it gives of R are periodic, and that there are finitely many translationequivalence classes of such tilings. The main result of the paper is that for any tiling of R by a bounded tile, any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling. This result implies a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups. 1. Introduction
Ergodic Theory On Moduli Spaces
 Ann. of Math
"... . Let M be a compact surface with Ø(M ) ! 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2)). Then the mapping class group \Gamma M of M acts on the moduli space X(M ) of flat Gbundles over M (possibly twisted by a fixed cen ..."
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Cited by 38 (5 self)
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. Let M be a compact surface with Ø(M ) ! 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2)). Then the mapping class group \Gamma M of M acts on the moduli space X(M ) of flat Gbundles over M (possibly twisted by a fixed central element of G). When M is closed, then \Gamma M preserves a symplectic structure on X(M ) which has finite total volume on M . More generally, the subspace of X(M ) corresponding to flat bundles with fixed behavior over @M carries a \Gamma M invariant symplectic structure. The main result is that \Gamma M acts ergodically on X(M ) with respect to the measure induced by the symplectic structure. Contents 1. Introduction 2 1.1. Statement of results 2 1.2. The NarasimhanSeshadri foliation of the universal moduli space 3 1.3. Moduli spaces over surfaces with boundary 4 1.4. Parabolic structures 5 1.5. Further speculations 5 1.6. Outline of proof 7 1.7. Acknowledgements 10 2. Preliminarie...
Spectrum of dynamical systems arising from Delone sets
 American Math. Soc.: Providence RI
, 1997
"... We investigate spectral properties of the translation action on the orbit closure of a Delone set. In particular, sufficient conditions for pure discrete spectrum are given, based on the notion of almost periodicity. Connections with diffraction spectrum are discussed. 1 Introduction A set ae R d ..."
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Cited by 37 (2 self)
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We investigate spectral properties of the translation action on the orbit closure of a Delone set. In particular, sufficient conditions for pure discrete spectrum are given, based on the notion of almost periodicity. Connections with diffraction spectrum are discussed. 1 Introduction A set ae R d is a Delone set if there exist positive constants R and r such that every ball of radius R intersects and every ball of radius r contains at most one point of . The collection of all such sets with fixed R and r can be equipped with a metric to form a compact space. The group R d acts on this space by translations. We study the spectral properties of this action restricted to some invariant subsets. We begin with a description of eigenvalues (with continuous eigenfunctions) assuming that the restricted action is minimal. Then we consider dynamics with respect to an ergodic invariant measure and obtain sufficient conditions for the system to have pure discrete spectrum. These conditions a...
A quantitative ergodic theory proof of Szemerédi’s theorem
, 2004
"... A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combinato ..."
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Cited by 34 (14 self)
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A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combinatorial proof using the Szemerédi regularity lemma and van der Waerden’s theorem, Furstenberg’s proof using ergodic theory, Gowers’ proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers’ more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, selfcontained version of this ergodic theory proof, and which is “elementary ” in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
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Cited by 34 (1 self)
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Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
The primes contain arbitrarily long polynomial progressions
 Acta Math
"... Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε suc ..."
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Cited by 30 (4 self)
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Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents
Geometric Aspects of Quantum Spin States
 COMMUN. MATH. PHYS.
, 1994
"... A number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S+l)invariant quantum spinS chains with the ..."
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Cited by 29 (14 self)
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A number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S+l)invariant quantum spinS chains with the interaction — P (0), where P (0) is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of Neel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the _p(θ) Spjn_s systems with the Potts and the FortuinKasteleyn randomcluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
"... ..."
The ergodic theoretical proof of Szemerédi’s theorem
 Bull. Amer. Math. Soc. (N.S
, 1982
"... some forty years earlier by Erdös and Turan: THEOREM I. Let A C Z be a subset of the integers of positive upper density, then A contains arbitrarily long arithmetic progressions. Partial results were obtained previously by K. F. Roth (1952) who established ..."
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Cited by 25 (1 self)
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some forty years earlier by Erdös and Turan: THEOREM I. Let A C Z be a subset of the integers of positive upper density, then A contains arbitrarily long arithmetic progressions. Partial results were obtained previously by K. F. Roth (1952) who established