On Lacunary Trigonometric Series.

1. Fundamental theorem. In a recent paper f I have proved the theorem that if a lacunary trigonometric series CO (1) X(a* cos nk6 + bk sin nk9) (nk+x/nk > q > 1, 0 ^ 0 ^ 2ir) 4-1 has its partial sums uniformly bounded on a set of 0 of positive measure, then the series (2) ¿(a*2 + bk2) k-l converges. The proof was based on the following lemma (which was not stated explicitly but is contained in the paper referred to, pp. 91-94). Lemma 1. Let E be an arbitrary measurable set of points of the interval (0, 2tt), m(E)>0. Then there exists a number N0 = No(q, E) such that, for N>No, we have r w (3) | ss-s " 0 \2dB 2= \m(E) £ (a*2 + bk2), Je k~N,+l where % denotes the Nth partial sum of the series (1), i.e. N stf = ^Ziak cos nk0 + bk sin nk6). Now we shall prove a somewhat more general theorem. Theorem 1. If the partial sums of the series (1) are uniformly bounded below on a set E of positive measure, then series (2) converges. If A is a positive constant, sufficiently large, we have Sn+A ^0 on E, and so (\sir\deg f i | sN + A | + A)d6 = 2AmiE) + fsNdd Je Je Je (4) = 2Am(E) + JZia^k + bkr¡k),