On the distribution of translates of additive functions

Our main purpose is to give necessary and sufficient conditions for the existence of the limit distribution of P ( E ) f ( n ) a ( x ) , where f is an additive function and P(E)f (n) = aof(n) + al f (n + 1) + . . . + akf(n + k). A sharp theorem can be derived from a recent result due to A. Hildebrand [5], if P(z) = ~, ajzJ satisfies the condition P(1) # 0 or P'(1) ~ 0. We consider the case ( z 1) 2 I P(z) (see Theorem 1). In Section 2 we shall investigate conditions under which