Representation of integers by near quadratic sequences

Following a statement of the well-known Erdo{double acute}s-Turán conjecture, Erdo{double acute}s mentioned the following even stronger conjecture: if the n-th term a_n of a sequence A of positive integers is bounded by αn², for some positive real constant α, 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 αn² (or from a quadratic polynomial with rational coefficients q(n)) by at most o(√log n), then the number of representations function is indeed unbounded.