Remarks on a Theorem of Zygmund

A well-known theorem of Zygmund (6) states that if n 1 < n 2 <. .. is a sequence of integers satisfying a (1) n~ +i/n~ > l+c (c > 0), k=1 converges for at least one x ; in fact the set of x for which (2) converges is of power c in any interval. Paley and Mary Weiss (5) extended this theorem for power series, i .e. (3) Y a i.znk k=1 converges for at least one z with I z I = 1 ; in fact the set of these z's is of power c on every arc. Kahane (3) calls a sequence of integers n 1 < n 2 <. .. a Zygmund sequence if whenever I a,; I > 0 the series (3) converges for at least one z with I z I = 1. Kennedy (4) proved that to every T,(k) > 0 (as k > o(there re is a sequence n l < n 2 <. .. for which n k+i/ni. > 1+9~ (k) and which is not a Zygmund sequence. Kennedy's result implies that in some sense Zygmund's theorem cannot be sharpened. Kahane (3) observes thatà slight change in the proof [of Kennedy] shows that a Zygmund sequence cannot contain arbitrarily long arithmetic progressions. Nothing more seems to be known .' I am going to prove the following