Erdős–Fuchs theorem

In mathematics, in the area of additive 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 too close of being a linear function.

The theorem is named after Paul Erdős and Wolfgang Heinrich Johannes Fuchs, who published it in 1956.

Let ${\textstyle {\mathcal {A}}\subseteq \mathbb {N} }$ be an infinite subset of the natural numbers and ${\textstyle r_{{\mathcal {A}},h}(n)}$ its representation function, which denotes the number of ways that a natural number ${\textstyle n}$ can be expressed as the sum of ${\textstyle h}$ elements of ${\textstyle {\mathcal {A}}}$ (taking order into account). We then consider the accumulated representation function:

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