Erdős–Fuchs theorem

In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.

The theorem is named after Paul Erdős and Wolfgang Heinrich Johannes Fuchs.

Statement
Let A be a subset of the natural numbers and r(n) denote the number of ways that a natural number n can be expressed as the sum of two elements of A (taking order into account). We consider the average


 * $$R(n) = (r(1)+r(2)+\cdots+r(n) ) / n . $$

The theorem states that


 * $$R(n) = C + O\left(n^{-3/4-\epsilon}\right) $$

cannot hold unless C=0.