Critical Numbers for Restricted Sumsets: Rigidity and Collapse in Finite Abelian Groups

This paper establishes a classification of the critical numbers for restricted sumsets in finite abelian groups, determining them exactly for even-order groups and bounding them for odd-order groups, while revealing a fundamental structural dichotomy governed by parity. For groups of even order, we prove a universal rigidity theorem: the index-$2$ subgroup creates an immutable arithmetic barrier at density $1/2$, fixing the critical number at $|G|/2+1$ regardless of the group’s internal structure. In sharp contrast, we demonstrate that for groups of odd order, this barrier vanishes, causing the critical threshold to collapse to significantly lower densities bounded by index-$5$ obstructions or the smallest prime divisor. These results unify and vastly generalize previous work on cyclic groups, providing a definitive structural theory for the transition from sparsity to saturation. As a decisive application, we resolve a conjecture of Han and Ren in algebraic coding theory. By translating the additive rigidity at density $1/2$ into a geometric constraint, we prove that for all sufficiently large $q$, any subset of rational points on an elliptic curve $E/\mathbb{F}_q$ generating an MDS code must satisfy the tight bound $|P|\le|E(\mathbb{F}_q)|/2$.

💡 Research Summary

The paper studies the problem of determining, for a finite abelian group G, the smallest size m such that every subset A⊆G with |A|≥m has its restricted k‑sumset Γₖ(A) equal to the whole group. The restricted k‑sumset is defined as the set of all sums of k pairwise distinct elements of A. The minimal such m is called the k‑critical number µₖ(G).

The authors first collect basic tools: a symmetry identity Γₖ(A)=A−Γ_{|A|−k}(A) (Lemma 1), a normalization lemma that allows one to translate A so that its densest fiber lies in the kernel of a homomorphism onto a prime‑order quotient (Lemma 2), and structural lemmas about the shape of large subsets (Lemma 3, Lemma 8). Lemma 3, based on work of Lev, says that for a sufficiently large group either |A|≤5/13·|G|, or A is contained in a coset of an index‑2 subgroup, or in a union of two cosets of an index‑5 subgroup, or else Γ₃(A)=G. Lemma 8 shows that distinct index‑2 cosets intersect in at most |G|/4 elements.

With these tools the paper proves two main theorems.

Theorem A (Rigidity). Let G be a finite abelian group of order g and let p(G) be its smallest prime divisor. Assume that g is large enough relative to its 2‑torsion (explicit bounds are given, e.g. g≥624·|G