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13
Measure and Dimension Functions: Measurability and Densities
 MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
, 1997
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Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
The structure of the σideal of σporous sets
, 1999
"... We show a general method of construction of nonσporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonσporous Suslin subset of a topologically complete metric space contains a nonσporous closed subset. We show also a sufficie ..."
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Cited by 6 (2 self)
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We show a general method of construction of nonσporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonσporous Suslin subset of a topologically complete metric space contains a nonσporous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a nonσporous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Hausdorff metric by K(E), then it is shown that each analytic subset of K(E) containing all countable compact subsets of E contains necessarily an element, which is nonσporous subset of E. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed nonσporous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the σideal of compact σporous sets.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
THE LIMITS OF DETERMINACY IN SECOND ORDER ARITHMETIC
, 2010
"... We establish the precise bounds for the amount of determinacy provable in second order arithmetic. We show that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many Π 0 3 classes, but it cannot prove that all finite Boolean combina ..."
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We establish the precise bounds for the amount of determinacy provable in second order arithmetic. We show that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many Π 0 3 classes, but it cannot prove that all finite Boolean combinations of Π 0 3 classes are determined. More specifically, we prove that Π 1 n+2CA 0 ⊢ nΠ 0 3DET, but that ∆ 1 n+2CA � nΠ 0 3DET, where nΠ 0 3 is the nth level in the difference hierarchy of Π 0 3 classes. We also show some conservativity results that imply that reversals for the theorems above are not possible. We prove that for every true Σ 1 4 sentence T (as for instance nΠ 0 3DET) and
Reducing subspaces
, 1993
"... Let T be a selfadjoint operator acting in a separable Hilbert space H. We establish a correspondence between the reducing subspaces of T that come from a spectral projection and the convex, normclosed bands in the set of finite Borel measures on R. If H is not separable, we still obtain a reducing ..."
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Let T be a selfadjoint operator acting in a separable Hilbert space H. We establish a correspondence between the reducing subspaces of T that come from a spectral projection and the convex, normclosed bands in the set of finite Borel measures on R. If H is not separable, we still obtain a reducing subspace corresponding to each convex normclosed band. These observations lead to a unified treatment of various reducing subspaces; moreover, they also settle some open questions and suggest new decompositions. 1 Reducing subspaces and bands Throughout this paper, we fix a selfadjoint operator T acting in Hilbert space H. As T is selfadjoint, it admits the representation T = � λ dE(λ) where E(·) R
On the complexity of some σideals of σPporous sets
 COMMENT.MATH.UNIV.CAROLIN. 44,3 (2003)531–554 531
, 2003
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Analytic representation of functions and a new quasianalyticity threshold
, 2006
"... We characterize precisely the possible rate of decay of the antianalytic half of a trigonometric series converging to zero almost everywhere. ..."
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We characterize precisely the possible rate of decay of the antianalytic half of a trigonometric series converging to zero almost everywhere.
The Structure Of The sigmaIdeal Of sigmaPorous Sets
, 1999
"... . We show a general method of construction of nonoeporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonoeporous Suslin subset of a topologically complete metric space contains a nonoeporous closed subset. We show also a suff ..."
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. We show a general method of construction of nonoeporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonoeporous Suslin subset of a topologically complete metric space contains a nonoeporous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a nonoeporous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Hausdorff metric by K(E), then it is shown that each analytic subset of K(E) containing all countable compact subsets of E contains necessarily an element, which is nonoeporous subset of E. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed nonoeporous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the oeideal of compact oeporous sets. 1. Introduction Let (P; ae) be a...
Contemporary Mathematics Uniqueness Questions for Multiple Trigonometric Series
"... Abstract. We survey some recent results on the uniqueness questions on multiple trigonometric series. Two basic questions, one about series which converges to zero and the other about the series which converge to an integrable function, are asked for four modes of convergence: unrestricted rectang ..."
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Abstract. We survey some recent results on the uniqueness questions on multiple trigonometric series. Two basic questions, one about series which converges to zero and the other about the series which converge to an integrable function, are asked for four modes of convergence: unrestricted rectangular convergence, spherical convergence, square convergence, and restricted rectangular convergence. We will either get into the details or outline some of the proofs for the known uniqueness theorems. Some results on the sets of uniqueness are also given. Finally, we will mention some interesting open questions in this area. Some of them are even onedimensional. We assume the reader has some basic knowledge of measure theory and Fourier analysis. Most of the topics and materials can be understood by upper level undergraduate