On the Complexity of Sets of Uniqueness and Extended
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Arxiv:1902.00426V3 [Math.CA] 4 Jan 2021 That the Empty Set ∅ Is a Set of Uniqueness and So [0, 1] Is a Set of Multiplicity
TRIGONOMETRIC SERIES AND SELF-SIMILAR SETS JIALUN LI AND TUOMAS SAHLSTEN Abstract. Let F be a self-similar set on R associated to contractions fj(x) = rjx + bj, j 2 A, for some finite A, such that F is not a singleton. We prove that if log ri= log rj is irrational for some i 6= j, then F is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of F . No separation conditions are assumed on F . We establish our result by showing that every self-similar measure µ on F is a Rajchman measure: the Fourier transform µb(ξ) ! 0 as jξj ! 1. The rate of µb(ξ) ! 0 is also shown to be logarithmic if log ri= log rj is diophantine for some i 6= j. The proof is based on quantitative renewal theorems for stopping times of random walks on R. 1. Introduction and the main result The uniqueness problem in Fourier analysis that goes back to Riemann [42] and Cantor [10] concerns the following question: suppose we have two converging trigonometric series P 2πinx P 2πinx ane and bne with coefficients an; bn 2 C such that for \many" x 2 [0; 1] they agree: X 2πinx X 2πinx ane = bne ; (1.1) n2Z n2Z then are the coefficients an = bn for all n 2 Z? For how \many" x 2 [0; 1] do we need to have (1.1) so that an = bn holds for all n 2 Z? If we assume (1.1) holds for all x 2 [0; 1], then using Toeplitz operators Cantor [10] proved that indeed an = bn for all n 2 Z. -
Topological Description of Riemannian Foliations with Dense Leaves
TOPOLOGICAL DESCRIPTION OF RIEMANNIAN FOLIATIONS WITH DENSE LEAVES J. A. AL¶ VAREZ LOPEZ*¶ AND A. CANDELy Contents Introduction 1 1. Local groups and local actions 2 2. Equicontinuous pseudogroups 5 3. Riemannian pseudogroups 9 4. Equicontinuous pseudogroups and Hilbert's 5th problem 10 5. A description of transitive, compactly generated, strongly equicontinuous and quasi-e®ective pseudogroups 14 6. Quasi-analyticity of pseudogroups 15 References 16 Introduction Riemannian foliations occupy an important place in geometry. An excellent survey is A. Haefliger’s Bourbaki seminar [6], and the book of P. Molino [13] is the standard reference for riemannian foliations. In one of the appendices to this book, E. Ghys proposes the problem of developing a theory of equicontinuous foliated spaces parallel- ing that of riemannian foliations; he uses the suggestive term \qualitative riemannian foliations" for such foliated spaces. In our previous paper [1], we discussed the structure of equicontinuous foliated spaces and, more generally, of equicontinuous pseudogroups of local homeomorphisms of topological spaces. This concept was di±cult to develop because of the nature of pseudogroups and the failure of having an in¯nitesimal characterization of local isome- tries, as one does have in the riemannian case. These di±culties give rise to two versions of equicontinuity: a weaker version seems to be more natural, but a stronger version is more useful to generalize topological properties of riemannian foliations. Another relevant property for this purpose is quasi-e®ectiveness, which is a generalization to pseudogroups of e®ectiveness for group actions. In the case of locally connected fo- liated spaces, quasi-e®ectiveness is equivalent to the quasi-analyticity introduced by Date: July 19, 2002. -
D^F(X) = Lim H~2[F(X + H) +F(X -H)- 2F{X)}
422 BOOK REVIEWS I enjoyed reading this book. While presenting a lot of mathematics, carefully and in considerable detail, Davis has managed to be conversational whenever possible, and to tell us where he's going and why. A class of advanced undergraduates with good linear algebra preparation should be able to read this book, to work the exercises, and to discuss their work together. Even though circulant matrices are themselves not of primary importance, their study, via this excellent book, could be a valuable part of discovering mathe matics as a discipline. DAVID H. CARLSON BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 7, Number 2, September 1982 © 1982 American Mathematical Society 0273-0979/82/0000-0227/$02.00 Essays in commutative harmonic analysis, by Colin C. Graham and O. Carruth McGehee, Grundlehren der Mathematischen Wissenschaften, Band 238, Springer-Verlag, New York, 1979, xxi + 464 pp., $42.00. Harmonic Analysis has passeed through a series of distinct epochs. The subject began in 1753 when Daniel Bernoulli gave a "general solution" to the problem of the vibrating string, previously investigated by d'Alembert and Euler, in terms of trigonometric series. This first epoch continues up to Fourier's book (1822). It is not completely clear what was going on in the minds of the harmonic analysts of this period, but the dominant theme seems to me to be the spectral theory of the operator d2/dx2 on T = R/Z. In 1829 Modern Analysis emerges fully armed from the head of Dirichlet [3]. His proof that the Fourier series of a monotone function converges pointwise everywhere meets the modern standards of rigor. -
Separation Problems and Forcing∗
Separation problems and forcing∗ Jindˇrich Zapletal y Academy of Sciences, Czech Republic University of Florida February 21, 2013 Abstract Certain separation problems in descriptive set theory correspond to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle T can be covered by @1 many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it. 1 Introduction Let X be a Polish space and K(X) its hyperspace of compact subsets of X. If J ⊆ I ⊆ K(X) are two collections of compact sets, one may ask whether it is possible to separate J from K(X) n I by an analytic set, i.e. if there is an analytic set A ⊆ K(X) such that J ⊆ A ⊆ I. A typical context arises when I;J are both coanalytic σ-ideals of compact sets or even J = K@0 (X), the collection of countable compact subsets of X. For example, Pelant and Zelen´y[9, Theorem 4.8] showed that K@0 (X) cannot be separated by an analytic set from the collection of non-σ-porous sets, where X is an arbitrary compact separable metric space. On the other hand, Loomis [7] [5, page 129 and 185] showed that K@0 (T) can be separated by an analytic set from the collection of compact sets of multiplicity, where T is the unit circle. In this paper, I explore the connection of such separation problems with the theory of definable proper forcing [13]. -
Dense Embeddings of Sigma-Compact, Nowhere Locallycompact Metric Spaces
proceedings of the american mathematical society Volume 95. Number 1, September 1985 DENSE EMBEDDINGS OF SIGMA-COMPACT, NOWHERE LOCALLYCOMPACT METRIC SPACES PHILIP L. BOWERS Abstract. It is proved that a connected complete separable ANR Z that satisfies the discrete n-cells property admits dense embeddings of every «-dimensional o-compact, nowhere locally compact metric space X(n e N U {0, oo}). More gener- ally, the collection of dense embeddings forms a dense Gs-subset of the collection of dense maps of X into Z. In particular, the collection of dense embeddings of an arbitrary o-compact, nowhere locally compact metric space into Hubert space forms such a dense Cs-subset. This generalizes and extends a result of Curtis [Cu,]. 0. Introduction. In [Cu,], D. W. Curtis constructs dense embeddings of a-compact, nowhere locally compact metric spaces into the separable Hilbert space l2. In this paper, we extend and generalize this result in two distinct ways: first, we show that any dense map of a a-compact, nowhere locally compact space into Hilbert space is strongly approximable by dense embeddings, and second, we extend this result to finite-dimensional analogs of Hilbert space. Specifically, we prove Theorem. Let Z be a complete separable ANR that satisfies the discrete n-cells property for some w e ÍV U {0, oo} and let X be a o-compact, nowhere locally compact metric space of dimension at most n. Then for every map f: X -» Z such that f(X) is dense in Z and for every open cover <%of Z, there exists an embedding g: X —>Z such that g(X) is dense in Z and g is °U-close to f. -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
Descriptive Set Theory
Descriptive Set Theory David Marker Fall 2002 Contents I Classical Descriptive Set Theory 2 1 Polish Spaces 2 2 Borel Sets 14 3 E®ective Descriptive Set Theory: The Arithmetic Hierarchy 27 4 Analytic Sets 34 5 Coanalytic Sets 43 6 Determinacy 54 7 Hyperarithmetic Sets 62 II Borel Equivalence Relations 73 1 8 ¦1-Equivalence Relations 73 9 Tame Borel Equivalence Relations 82 10 Countable Borel Equivalence Relations 87 11 Hyper¯nite Equivalence Relations 92 1 These are informal notes for a course in Descriptive Set Theory given at the University of Illinois at Chicago in Fall 2002. While I hope to give a fairly broad survey of the subject we will be concentrating on problems about group actions, particularly those motivated by Vaught's conjecture. Kechris' Classical Descriptive Set Theory is the main reference for these notes. Notation: If A is a set, A<! is the set of all ¯nite sequences from A. Suppose <! σ = (a0; : : : ; am) 2 A and b 2 A. Then σ b is the sequence (a0; : : : ; am; b). We let ; denote the empty sequence. If σ 2 A<!, then jσj is the length of σ. If f : N ! A, then fjn is the sequence (f(0); : : :b; f(n ¡ 1)). If X is any set, P(X), the power set of X is the set of all subsets X. If X is a metric space, x 2 X and ² > 0, then B²(x) = fy 2 X : d(x; y) < ²g is the open ball of radius ² around x. Part I Classical Descriptive Set Theory 1 Polish Spaces De¯nition 1.1 Let X be a topological space. -
Completeness Properties of the Generalized Compact-Open Topology on Partial Functions with Closed Domains
Topology and its Applications 110 (2001) 303–321 Completeness properties of the generalized compact-open topology on partial functions with closed domains L’ubica Holá a, László Zsilinszky b;∗ a Academy of Sciences, Institute of Mathematics, Štefánikova 49, 81473 Bratislava, Slovakia b Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA Received 4 June 1999; received in revised form 20 August 1999 Abstract The primary goal of the paper is to investigate the Baire property and weak α-favorability for the generalized compact-open topology τC on the space P of continuous partial functions f : A ! Y with a closed domain A ⊂ X. Various sufficient and necessary conditions are given. It is shown, e.g., that .P,τC/ is weakly α-favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α-favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τC, the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results. 2001 Elsevier Science B.V. All rights reserved. Keywords: Generalized compact-open topology; Fell topology; Compact-open topology; Topological games; Banach–Mazur game; Weakly α-favorable spaces; Baire spaces; Graph topology; π-base; Restriction mapping AMS classification: Primary 54C35, Secondary 54B20; 90D44; 54E52 0. -
Retracing Cantor's First Steps in Brouwer's Company
RETRACING CANTOR’S FIRST STEPS IN BROUWER’S COMPANY WIM VELDMAN What we call the beginning is often the end And to make an end is to make a beginning. The end is where we start from. T.S. Eliot, Little Gidding, 1942 Abstract. We prove intuitionistic versions of the classical theorems saying that all countable closed subsets of [−π,π] and even all countable subsets of [−π,π] are sets of uniqueness. We introduce the co-derivative of an open subset of the set R of the real numbers as a constructively possibly more useful notion than the derivative of a closed subset of R. We also have a look at an intuitionistic version of Cantor’s theorem that a closed set is the union of a perfect set and an at most countable set. 1. Introduction G. Cantor discovered the transfinite and the uncountable while studying and extending B. Riemann’s work on trigonometric series. He then began set theory and forgot his early problems, see [17] and [22]. Cantor’s work fascinated L.E.J. Brouwer but he did not come to terms with it and started intuitionistic mathematics. Like Shakespeare, who wrote his plays as new versions of works by earlier playwrights, he hoped to turn Cantor’s tale into a better story. Brouwer never returned to the questions and results that caused the creation of set theory. Adopting Brouwer’s point of view, we do so now. Brouwer insisted upon a constructive interpretation of statements of the form A ∨ B and ∃x ∈ V [A(x)]. -
Set Ideal Topological Spaces
Set Ideal Topological Spaces W. B. Vasantha Kandasamy Florentin Smarandache ZIP PUBLISHING Ohio 2012 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com Copyright 2012 by Zip Publishing and the Authors Peer reviewers: Prof. Catalin Barbu, V. Alecsandri National College, Mathematics Department, Bacau, Romania. Prof. Valeri Kroumov, Okayama Univ. of Science, Japan. Dr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052, USA. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-193-3 EAN: 9781599731933 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET IDEALS IN RINGS 9 Chapter Three SET IDEAL TOPOLOGICAL SPACES 35 Chapter Four NEW CLASSES OF SET IDEAL TOPOLOGICAL SPACES AND APPLICATIONS 93 3 FURTHER READING 109 INDEX 111 ABOUT THE AUTHORS 114 4 PREFACE In this book the authors for the first time introduce a new type of topological spaces called the set ideal topological spaces using rings or semigroups, or used in the mutually exclusive sense. This type of topological spaces use the class of set ideals of a ring (semigroups). The rings or semigroups can be finite or infinite order. By this method we get complex modulo finite integer set ideal topological spaces using finite complex modulo integer rings or finite complex modulo integer semigroups. Also authors construct neutrosophic set ideal toplogical spaces of both finite and infinite order as well as complex neutrosophic set ideal topological spaces. -
ON BERNSTEIN SETS Let Us Recall That a Subset of a Topological Space
ON BERNSTEIN SETS JACEK CICHON´ ABSTRACT. In this note I show a construction of Bernstein subsets of the real line which gives much more information about the structure of the real line than the classical one. 1. BASIC DEFINITIONS Let us recall that a subset of a topological space is a perfect set if is closed set and contains no isolated points. Perfect subsets of real line R have cardinality continuum. In fact every perfect set contains a copy of a Cantor set. This can be proved rather easilly by constructing a binary tree of decreasing small closed sets. There are continnum many perfect sets. This follows from a more general result: there are continuum many Borel subsets of real. We denote by R the real line. We will work in the theory ZFC. Definition 1. A subset B of the real line R is a Bernstein set if for every perfect subset P of R we have (P \ B 6= ;) ^ (P n B 6= ;) : Bernstein sets are interesting object of study because they are Lebesgue nonmeasurable and they do not have the property of Baire. The classical construction of a Berstein set goes as follows: we fix an enumeration (Pα)α<c of all perfect sets, we define a transfinite sequences (pα), (qα) of pairwise different points such that fpα; qαg ⊆ Pα, we put B = fpα : α < cg and we show that B is a Bernstein set. This construction can be found in many classical books. The of this note is to give slightly different construction, which will generate simultanously a big family of Bermstein sets.We start with one simple but beautifull result: Lemma 1. -
Lecture 8: the Real Line, Part II February 11, 2009
Lecture 8: The Real Line, Part II February 11, 2009 Definition 6.13. a ∈ X is isolated in X iff there is an open interval I for which X ∩ I = {a}. Otherwise, a is a limit point. Remark. Another way to state this is that a is isolated if it is not a limit point of X. Definition 6.14. X is a perfect set iff X is closed and has no isolated points. Remark. This definition sounds nice and tidy, but there are some very strange perfect sets. For example, the Cantor set is perfect, despite being nowhere dense! Our goal will be to prove the Cantor-Bendixson theorem, i.e. the perfect set theorem for closed sets, that every closed uncountable set has a perfect subset. Lemma 6.15. If P is a perfect set and I is an open interval on R such that I ∩ P 6= ∅, then there exist disjoint closed intervals J0,J1 ⊂ I such that int[J0] ∩ P 6= ∅ and int[J1] ∩ P 6= ∅. Moreover, we can pick J0 and J1 such that their lengths are both less than any ² > 0. Proof. Since P has no isolated points, there must be at least two points a0, a1 ∈ I ∩ P . Then just pick J0 3 a0 and J1 3 a1 to be small enough. SDG Lemma 6.16. If P is a nonempty perfect set, then P ∼ R. Proof. We exhibit a one-to-one mapping G : 2ω → P . Note that 2ω can be viewed as the set of all infinite paths in a full, infinite binary tree with each edge labeled by 0 or 1.