Representation functions of sequences in additive number theory

Abstract:

Let $\mathcal {A}$ be a set of nonnegative integers, and let $r_2^\mathcal {A}(n)$ denote the number of representations of n in the form $n = {a_i} + {a_j}$ with ${a_i},{a_j} \in \mathcal {A}$. The set $\mathcal {A}$ is periodic if $a \in \mathcal {A}$ implies $a + m \in \mathcal {A}$ for some $m \geqslant 1$ and all $a > N$. It is proved that if $\mathcal {A}$ is not periodic, then for every set $\mathcal {B} \ne \mathcal {A}$ there exist infinitely many n such that $r_2^\mathcal {A}(n) \ne r_2^\mathcal {B}(n)$. Moreover, all pairs of periodic sets $\mathcal {A}$ and $\mathcal {B}$ are constructed that satisfy $r_2^\mathcal {A}(n) = r_2^\mathcal {B}(n)$ for all but finitely many n.