We show that if {a(k)}(k) is bounded then [GRAPHICS] for almost every 0 less than or equal to x (l)ess than or equal to 1 where x = [GRAPHICS] is the dyadic expansion of x. It is also shown that (1/n) Sigma(k=1)(n) a(k) exp (2 pi i . p(k)x) --> 0 almost everywhere where p > 1 is any fixed integer.