The Erdös-Fuchs theorem on the square of a power series

Erdös and Fuchs proved that if a₁, a₂,… is a sequence of nonnegative integers and R(n) is the number of ordered pairs (i, j) with a_i + a_j ≤ n, then it is impossible to have $\text{R(n) = An + o(n}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{log}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{n)}$ as n → + ∞, for some positive constant A. This paper gives a generalization of this result, in which An is replaced by a function of n whose second differences are nonnegative from some point on.