On additive representation functions of finite sets, I (Variation)

Abstract

If m ∈ ℕ, ℤ _m is the additive group of the modulo m residue classes, $\mathcal{A} \subset \mathbb{Z}_m$ and n ∈ ℕ, ℤ _m , then let $R\left( {\mathcal{A},n} \right)$ denote the number of solutions of a+a′ = n with $a,a' \in \mathcal{A}$. The variation $V(\mathcal{A}) = \mathop {\max }\limits_{n \in \mathbb{Z}_m } |R(\mathcal{A},n + 1) - R(\mathcal{A},n)|$ is estimated in terms of the number of a’s with $a - 1 \notin \mathcal{A}$, $a \in \mathcal{A}$.