This paper shows for a bounded tile that all tilings it gives of R are periodic, and that there are finitely many translation-equivalence 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

It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigma-compact 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.

. 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 G-bundles 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 Narasimhan-Seshadri 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...

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, self-contained 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.

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...

Abstract. We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that the crossed product of a stably finite simple unital C*-algebra with tracial rank zero by an action with this property again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Izumi. We use these results, together with work of Elliott-Evans and Kishimoto, to give an inductive proof that every simple higher dimensional noncommutative torus is an AT algebra. We further prove that the crossed product of every simple higher dimensional noncommutative torus by the flip is an AF algebra, and that the crossed products of irrational rotation algebras by the standard actions of Z3, Z4, and Z6 are simple AH algebras with real rank zero. In the

Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued 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

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 spin-S 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 Fortuin-Kasteleyn random-cluster 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

Abstract. There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms U d (G), d = 1, 2, 3,... on a finite additive group G; in particular, to detect arithmetic progressions of length k in G it is important to know under what circumstances the U k−1 (G) norm can be large. The U 1 (G) norm is trivial, and the U 2 (G) norm can be easily described in terms of the Fourier transform. In this paper we systematically study the U 3 (G) norm, defined for any function f: G → C on a finite additive group G by the formula

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