When Do Restricted Sumsets Cover a Finite Abelian Group?

For a subset $A$ of a finite abelian group $G$, let ${\mathrm{\Sigma }}_k(A)$ denote the set of sums of $k$ distinct elements of $A$, and let $\mathrm{\Sigma }(A)={\bigcup }_{k\ge 1}{\mathrm{\Sigma }}_k(A)$. The problem of determining when $\mathrm{\Sigma }(A)=G$ goes back to Erdős and Heilbronn. Nguyen and Vu later formulated the corresponding fixed-length question: for a prescribed $k$, when does ${\mathrm{\Sigma }}_k(A)=G$? For groups of odd order, we prove the sharp large-order threshold $\mathrm{\mid }A\mathrm{\mid }>\mathrm{\mid }G\mathrm{\mid }\mathrm{/}3+1$, which forces ${\mathrm{\Sigma }}_k(A)=G$ throughout the entire middle range $4\le k\le \mathrm{\mid }A\mathrm{\mid }-4$. This determines the limiting thresholds posed by Girard, Griffiths and Hamidoune. If $p$ is the least prime divisor of $\mathrm{\mid }G\mathrm{\mid }$, then, for fixed odd $p$ and sufficiently large $\mathrm{\mid }G\mathrm{\mid }$, the exact threshold $\mathrm{\mid }A\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }\mathrm{/}p+p-1$ forces covering throughout the middle range beginning at $2p-2$. Thus, throughout this range, prescribing the number of summands does not change the sharp covering threshold. For $3\le k\le 2p-3$, the restricted and ordinary $k$-fold covering problems have the same asymptotic threshold. As an application to MDS elliptic codes on an elliptic curve $E\mathrm{/}{\text{F}}_q$, we sharpen the half-scale conjecture of Li, Wan and Zhang, obtaining the sharp $1\mathrm{/}3$-scale bound when $\mathrm{\mid }E({\text{F}}_q)\mathrm{\mid }$ is odd and, more generally, the exact bound determined by the least prime divisor of $\mathrm{\mid }E({\text{F}}_q)\mathrm{\mid }$.