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

Bocong Chen, Jing Huang

TL;DR

The paper classifies the critical thresholds for restricted sumsets in finite abelian groups, showing a parity-driven dichotomy: even-order groups exhibit a universal rigidity at density $1/2$ (fixed critical number $\mu_k(G)=\tfrac{|G|}{2}+1$ for a broad range of $k$), while odd-order groups admit substantially lower thresholds governed by index-$5$ obstructions or the smallest prime divisor. The core method combines symmetry and normalization for sumsets, projection to prime quotients, and inverse-structure results (including the Dias da Silva–Hamidoune theorem, Lev’s lemma, and the Devos–Goddyn–Mohar framework) to establish density-based covering results and lift them to general $G$. The two main theorems (A and B) yield a complete rigidity-collapse picture, with immediate consequences for zero-sum and coding theory problems, including resolving a conjecture of Han and Ren by showing that for sufficiently large $q$, any MDS elliptic code must satisfy $|P|\\le |E(\mathbb{F}_q)|/2$. The work unifies and extends prior cyclic-group results to arbitrary finite abelian groups, providing a structural theory of the transition from sparsity to saturation in restricted sumsets.

Abstract

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 $|P|\le|E(\mathbb{F}_q)|/2$.

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

TL;DR

The paper classifies the critical thresholds for restricted sumsets in finite abelian groups, showing a parity-driven dichotomy: even-order groups exhibit a universal rigidity at density (fixed critical number for a broad range of ), while odd-order groups admit substantially lower thresholds governed by index- obstructions or the smallest prime divisor. The core method combines symmetry and normalization for sumsets, projection to prime quotients, and inverse-structure results (including the Dias da Silva–Hamidoune theorem, Lev’s lemma, and the Devos–Goddyn–Mohar framework) to establish density-based covering results and lift them to general . The two main theorems (A and B) yield a complete rigidity-collapse picture, with immediate consequences for zero-sum and coding theory problems, including resolving a conjecture of Han and Ren by showing that for sufficiently large , any MDS elliptic code must satisfy . The work unifies and extends prior cyclic-group results to arbitrary finite abelian groups, providing a structural theory of the transition from sparsity to saturation in restricted sumsets.

Abstract

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- subgroup creates an immutable arithmetic barrier at density , fixing the critical number at 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- 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 into a geometric constraint, we prove that for all sufficiently large , any subset of rational points on an elliptic curve generating an MDS code must satisfy the tight bound .
Paper Structure (12 sections, 31 theorems, 120 equations)

This paper contains 12 sections, 31 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Theorems A and B
  4. Proof of Theorem A
  5. Case 1: $a_1 \le k-3$.
  6. Case 2: $a_1 > k-3$.
  7. Case 1: $\min(a_1, a_2) \ge 2$.
  8. Case 2: $\min(a_1, a_2) = 1$.
  9. Case 1: $u\ge k-3$.
  10. Case 2: $u < k-3$.
  11. Proof of Theorem B
  12. Proof of the Conjecture on MDS Elliptic Codes

Key Result

Lemma 1

Let $G$ be a finite abelian group and let $A \subseteq G$ be a subset of size $a$. Let $\overline{A} = \sum_{\alpha \in A} \alpha$ denote the sum of the elements of $A$. Then, for every integer $k$ satisfying $0 \le k \le a$, we have the identity Consequently, $\Gamma_k(A) = G$ if and only if $\Gamma_{a-k}(A) = G$. In particular, if $\Gamma_k(A) = G$ for all $1 \le k \le \lfloor a/2 \rfloor$, the

Theorems & Definitions (60)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Corollary 5
  • proof
  • Lemma 6: DGM theorem
  • Corollary 7
  • ...and 50 more