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

Let A={a1,a2,⋯}(a1<a2<⋯) be an infinite sequence of nonnegative integers, and let RA,2(n) denote the number of solutions of ax+ay=n(ax,ay∈A). P. Erdős, A. Sárközy and V. T. Sós proved that if limN→∞B(A,N)N=+∞ then |Δ1(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 Δl denotes the l-th difference. Their result was extended to Δl(RA,2(n)) for any fixed l≥2. In this paper we give further generalizations of this problem. [ABSTRACT FROM AUTHOR]