Hereditary properties of the class of closed sets of uniqueness for trigonometric series

It is shown that theσ-idealU ₀ of closed sets of extended uniqueness inT is hereditarily non-Borel, i.e. every “non-trivial”σ-ideal of closed setsI⊆U ₀ is non-Borel. This implies both the result of Solovay, Kaufman that bothU ₀ andU (theσ-ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset ofT of extended uniqueness is of the first category. A further extension to ideals contained inU ₀ is given. Research partially supported by NSF Grant DMS-8718847.