Kechris, Alexander S. Hereditary properties of the class of closed sets of uniqueness for trigonometric series. (English) Zbl 0769.42006 Isr. J. Math. 73, No. 2, 189-198 (1991). Summary: It is shown that the \(\sigma\)-ideal \(U_ 0\) of closed sets of extended uniqueness in \(\mathbb{T}\), \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\) the circle group, is hereditarily non-Borel, i.e., every “nontrivial” \(\sigma\)-ideal of closed sets \(I\subseteq U_ 0\) is non-Borel. This implies both the result of Solovay and Kaufman that both \(U_ 0\) and \(U\) (the \(\sigma\)- ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset of \(\mathbb{T}\) of extended uniqueness is of the first category. A further extension to ideals contained in \(U_ 0\) is given. Cited in 1 ReviewCited in 11 Documents MSC: 42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization 03E15 Descriptive set theory 28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) Keywords:hereditary properties; closed sets of uniqueness for trigonometric series; Rajchman measure; theorem of Debs-Saint Raymond PDFBibTeX XMLCite \textit{A. S. Kechris}, Isr. J. Math. 73, No. 2, 189--198 (1991; Zbl 0769.42006) Full Text: DOI References: [1] Bary, N., Sur l’unicité du développement trigonometrique, Fund. Math., 9, 62-115 (1927) · JFM 53.0261.01 [2] Debs, G.; Saint Raymond, J., Ensembles d’unicité et d’unicité au sens large, Ann. Inst. Fourier Grenoble, 37, 3, 217-239 (1987) · Zbl 0618.42004 [3] Graham, C. C.; McGehee, O. C., Essays in Commutative Harmonic Analysis, Grund. Math. Wissen., Vol. 238 (1979), New York: Springer-Verlag, New York · Zbl 0439.43001 [4] Kaufman, R., Fourier transforms and descriptive set theory, Mathematika, 31, 336-339 (1984) · Zbl 0604.42009 · doi:10.1112/S0025579300012547 [5] Kaufman, R., Absolutely convergent Fourier series and some classes of sets, Bull. Sci. Math., 109, 363-372 (1985) · Zbl 0608.42007 [6] A. S. Kechris and A. Louveau,Descriptive set theory and the structure of sets of uniqueness, London Math. Soc. Lecture Note Ser., 128, Cambridge University Press, 1987. · Zbl 0642.42014 [7] Kechris, A. S.; Louveau, A.; Woodin, W. H., The structure of σ-ideals of compact sets, Trans. Am. Math. Soc., 301, 1, 263-288 (1987) · Zbl 0633.03043 · doi:10.2307/2000338 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.