Representation of Integers by Near Quadratic Sequences

Following a statement of the well-known Erdos-Turan conjecture, Erdos mentioned the following even stronger conjecture: if the n-th term a(n) of a sequence A of positive integers is bounded by alpha n(2), for some positive real constant alpha, then the number of representations of n as a sum of two terms from A is an unbounded function of n. Here we show that if a(n) differs from alpha n(2) (or from a quadratic polynomial with rational coefficients q(n)) by at most o(root logn), then the number of representations function is indeed unbounded.