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We study the family lSo of the sets on which some series of the form EkCN Isinlrnkxl is uniformly bounded. We show that the families 8' of all sets admitting the boundary c form a hierarchy which is incontinuous with respect to the operations of intersection and union. Let us recall two kinds of thin sets studied in harmonic analysis. A set X C [0,1] is called an N-set (in honour of V. V. Nemytskil) if there exists a trigonometric series absolutely converging on X but not converging absolutely everywhere; it is called an No -set if there exists a series of the form EkeN Isin 7rnk X converging on X. J. Arbault [1] showed that the family fib of all No-sets is a proper subfamily of the family XV of all N-sets, and both families share some common properties. In the paper [3] we examined several modifications of the definitions of the families XV and gO, and compared the obtained families one to another. We showed that the family Bo of all sets on which some series of the form EkeN 1sinlrnkX1 is uniformly bounded differs from previously known "classical" families. In the present paper we continue the study of the family Bo considering the hierarchy of the families l3o of all sets admitting the boundary c. We show that if c c l3d for every c > 0. For a review of families of trigonometric thin sets, some historical notes, and also for many new results we refer the reader to the paper [2]. We shall deal with the quotient group ?7 = IR/Z; however, we will not distinguish between the elements of ?7 and the reals, or between the functions defined on ?7 and the corresponding periodic functions on JR. For a real x, let [x] denote the integer part of x and let 11x11 denote the distance of x to the nearest integer, i.e. [x] = max{k E 2: k < x} and 11x11 = min{Ix-k1: k E 2}. Let us note that 11x11 -Ilyl < jIX + Y11 < ||XII + Ilyjl and 11-x11 = 11x11 for all x, y E T. The space ? equipped with the metric Q(x, y) = 1X y1 is a compact topological group. In an accordance with [3], we define Bo-sets as follows. Definition 1. A set X C ?7 is a Bo-set if there exists an increasing sequence {nk}kGN of natural numbers such that the series EkeN 11nkX11 is uniformly bounded Received by the editors January 12, 1999. 2000 Mathematics Subject Classification. Primary 43A46; Secondary 42A05, 42A32.

A hierarchy of thin sets related to the boundedness of trigonometric series