Let a1<a2<... be an infinite sequence of positive integers, let k≥2 be a fixed integer and denote by Rk(n) the number of solutions of n=ai1+ai2+...+aik. P. Erdős and A. Srkzy proved that if F(n) is a monotonic increasing arithmetic function with F(n)→+∞ and F(n)=o(n(log
n)-2) then |R2(n)-F(n)| =o((F(n))1/2) cannot hold. The aim of this paper is to extend this result to k>2.
Montgomery and Vaughan improved a theorem of Erdős and Fuchs for an arbitrary sequence. Srkzy extended this theorem of Erdős
and Fuchs for two arbitrary sequences which are "near" in a certain sense. Using the idea of Jurkat (differentiation of the
generating function), we will extend similarly the result of Montgomery and Vaughan for "sufficiently near" sequences.
Let 0 ≦ a1 < a2 < ⋯ be an infinite sequence of integers and let r1(A, n) = |(i;j): ai + aj = n, i ≦ j|. We show that if d > 0 is an integer, then there does not exist n0 such that d ≦ r1 (A, n) ≦ d + [√2d + ] for n > n0.