When Do Restricted Sumsets Cover a Finite Abelian Group?
Abstract
For a subset A of a finite abelian group G, let Σk(A) denote the set of sums of k distinct elements of A, and let Σ(A)=⋃k≥1Σk(A). The problem of determining when Σ(A)=G goes back to Erdős and Heilbronn. Nguyen and Vu later formulated the corresponding fixed-length question: for a prescribed k, when does Σk(A)=G? For groups of odd order, we prove the sharp large-order threshold |A|>|G|/3+1, which forces Σk(A)=G throughout the entire middle range 4≤ k≤ |A|−4. This determines the limiting thresholds posed by Girard, Griffiths and Hamidoune. If p is the least prime divisor of |G|, then, for fixed odd p and sufficiently large |G|, the exact threshold |A|≥ |G|/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≤ k≤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/Fq, we sharpen the half-scale conjecture of Li, Wan and Zhang, obtaining the sharp 1/3-scale bound when |E(Fq)| is odd and, more generally, the exact bound determined by the least prime divisor of |E(Fq)|.
Related
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BibTeX
@misc{han2026when,
title = {{When Do Restricted Sumsets Cover a Finite Abelian Group?}},
author = {Dongchun Han and Yuan Ren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14496},
eprint = {2610.14496},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14496},
}
RIS
TY - UNPB
TI - When Do Restricted Sumsets Cover a Finite Abelian Group?
AU - Dongchun Han
AU - Yuan Ren
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14496
UR - https://arxiv.org/abs/2610.14496
AB - For a subset A of a finite abelian group G, let Σ_k(A) denote the set of sums of k distinct elements of A, and let Σ(A)=⋃_k≥1Σ_k(A). The problem of determining when Σ(A)=G goes back to Erdős and Heilbronn. Nguyen and Vu later formulated the corresponding fixed-length question: for a prescribed k, when does Σ_k(A)=G? For groups of odd order, we prove the sharp large-order threshold |A|>|G|/3+1, which forces Σ_k(A)=G throughout the entire middle range 4≤ k≤ |A|−4. This determines the limiting thresholds posed by Girard, Griffiths and Hamidoune. If p is the least prime divisor of |G|, then, for fixed odd p and sufficiently large |G|, the exact threshold |A|≥ |G|/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≤ k≤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/F_q, we sharpen the half-scale conjecture of Li, Wan and Zhang, obtaining the sharp 1/3-scale bound when |E(F_q)| is odd and, more generally, the exact bound determined by the least prime divisor of |E(F_q)|.
ER -