Generalizations of some results about the regularity properties of an additive representation function

摘要

Let A={a1,a2,}(a1a2) be an infinite sequence of nonnegative integers, and let RA,2(n) denote the number of solutions of ax+ay=n(ax,ayA). P. Erds, A. Sarkozy and V. T. Sos proved that if limNmml:mfracB(A,N)then |1(RA,2(n))| cannot be bounded, where B(A,N) denotes the number of blocks formed by consecutive integers in A up to N and l denotes the l-th difference. Their result was extended to l(RA,2(n)) for any fixed l2. In this paper we give further generalizations of this problem.