## Descriptive Set Theory and the Structure of Sets of UniquenessThe authors present some surprising connections that sets of uniqueness for trigonometic series have with descriptive set theory. They present many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. Topics covered include symmetric perfect sets and the solution to the Borel Basis Problem for U, the class of sets of uniqueness. To make the material accessible to both logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Because the book is essentially selfcontained and requires the minimum prerequisites, it will serve as an excellent introduction to the subject for graduate students and research workers in set theory and analysis. |

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### Contents

III | 21 |

V | 23 |

VI | 25 |

VII | 33 |

VIII | 41 |

IX | 46 |

X | 51 |

XI | 56 |

XXVIII | 174 |

XXIX | 182 |

XXX | 193 |

XXXII | 211 |

XXXIII | 220 |

XXXIV | 231 |

XXXVI | 239 |

XXXVII | 243 |

### Common terms and phrases

a-ideal assume Baire Category Theorem Banach space Bj(X Borel basis Borel sets Chapter Characterization Problem choose clearly closed subsets cnetnx compact construct continuous functions contradiction converges convex Corollary countable closed sets countable ordinal countable set countable unions Debs-Saint Raymond define Definition denote dense descriptive set theory extended uniqueness fact finite H-set Helson hence induction integer interior uniqueness Kaufman Kechris Kechris-Louveau Lemma linear Lyons M-set metrizable space Moreover nbhd norm open interval open set perfect sets Piatetski-Shapiro rank Pisot number PM(E Polish space positive measure PROB(E probability measure proof of Theorem property of Baire Proposition prove pseudofunction pseudomeasure Rajchman measure result sequence set E C T set of multiplicity set of synthesis sets of uniqueness Solovay subspace supp(S topology trigonometric series U-sets w*-closed w*-dense weak*-topology