On additive bases of finite abelian groups

Abstract

Let $G$ be a finite abelian group, let $p$ be the smallest prime divisor of $|G|$, and let $c_0(G)$ be the least integer $t$ such that every regular sequence over $G$ of length at least $t$ is an additive basis. Additive bases have been extensively studied in additive combinatorics, and the exact value of $c_0(G)$ has been determined for many groups of rank at least three and some groups of rank two (in which case groups of the form $C_p\oplus C_{pn}$ turn out to be exceptional). In this paper, we continue the study of $c_0(G)$ and obtain the following results. (1) Together with known results, we prove that $c_0(G)=|G|/p+p-2$ for all abelian groups $G$ with $\text{rank}(G)\geq 3$. (2) For distinct odd primes $p<q$, we determine \[ c_0(C_p\oplus C_{pq})=pq+2p-3,\qquad c_0(C_q\oplus C_{pq})=q^2+p-2. \] (3) Let $p$ be an odd prime, let $2\leq a\leq b$, and let $n$ be a positive integer with no prime divisor smaller than $p$. Then \[ c_0(C_{p^a}\oplus C_{p^bn})=p^{a+b-1}n+p-2. \] (4) For every odd integer $n\geq 3$, we prove \[ c_0(C_3\oplus C_{3n})=3n+3. \] Therefore, $c_0(G)$ is determined except possibly for groups of the form $C_p\oplus C_{pn}$ with $p\geq 5$.