[Paper Review] More on the $h$-critical numbers of finite abelian groups

Béla Bajnok|arXiv (Cornell University)|Nov 22, 2016

Limits and Structures in Graph Theory7 references3 citations

TL;DR

This paper fully determines the $\widehat{\chi}(G,h)$ and $\widehat{\chi}(G,[0,s])$ invariants for all finite abelian groups $G$, resolving two open problems in additive combinatorics. It proves that $\widehat{\chi}(G,h) = v(n,h) + 1$ and $\widehat{\chi}(G,[0,s]) = v(n,s) + 1$, where $v(n,h)$ is the maximum of $f_d(n,h) = \left(\left\lfloor\frac{d-2}{h}\right\rfloor + 1\right)\cdot\frac{n}{d}$ over divisors $d$ of $n$, extending known results on critical numbers to generating sets.

ABSTRACT

For a finite abelian group $G$, a nonempty subset $A$ of $G$, and a positive integer $h$, we let $hA$ denote the $h$-fold sumset of $A$; that is, $hA$ is the collection of sums of $h$ not-necessarily-distinct elements of $A$. Furthermore, for a positive integer $s$, we set $[0,s] A=\cup_{h=0}^s h A$. We say that $A$ is a generating set of $G$ if there is a positive integer $s$ for which $[0,s] A=G$. The $h$-critical number $χ(G,h)$ of $G$ is defined as the smallest positive integer $m$ for which $hA=G$ holds for every $m$-subset $A$ of $G$; similarly, $χ(G,[0,s])$ is the smallest positive integer $m$ for which $[0,s]A=G$ holds for every $m$-subset $A$ of $G$. We define $\widehatχ (G, h)$ as the smallest positive integer $m$ for which $hA=G$ holds for every generating $m$-subset $A$ of $G$; $\widehatχ (G, [0,s])$ is defined similarly. The value of $χ(G,h)$ has been determined by this author for all $G$ and $h$, and $\widehatχ (G, [0,s])$ was introduced and resolved for some special cases by Klopsch and Lev. Here we determine the remaining two quantities in all cases.

Motivation & Objective

• To determine the $\widehat{\chi}(G,h)$ and $\widehat{\chi}(G,[0,s])$ invariants for all finite abelian groups $G$, which measure the minimal size of a generating subset $A$ such that $hA = G$ or $[0,s]A = G$.
• To resolve two previously open problems in additive combinatorics concerning $h$-critical numbers when restricted to generating subsets.
• To extend the known theory of critical numbers—previously established for arbitrary subsets—to the more refined setting of generating subsets, capturing the minimal size required for sumset coverage in the context of group generation.
• To provide a complete characterization of these invariants using a function $v(n,h)$ derived from the divisors of the group order $n$, ensuring the results are uniform across all abelian groups.

Proposed method

• Define $v(n,h) = \max_{d \mid n} \left\{ \left(\left\lfloor\frac{d-2}{h}\right\rfloor + 1\right) \cdot \frac{n}{d} \right\}$, which captures the maximal size of an $h$-incomplete subset in a group of order $n$.
• Use induction on the group order $n$, distinguishing cases based on whether the maximum in $v(n,h)$ is attained at a proper divisor $d_0 < n$ or at $d = n$.
• For the case where $v(n,h)$ is attained at a proper divisor $d_0$, construct a preimage $A = \pi^{-1}(B)$ of a subset $B$ in the quotient group $G/H$ of order $d_0$, where $B$ is $h$-incomplete but generates $G/H$, ensuring $A$ generates $G$ but $hA \neq G$.
• For the case where $v(n,h)$ is attained at $d = n$, consider $G$ cyclic of prime order $p$, and construct $A = \{1, 2, \dots, \left\lfloor\frac{p-2}{h}\right\rfloor + 1\}$, showing $hA \neq \mathbb{Z}_p$ but $\langle A \rangle = \mathbb{Z}_p$.
• For composite $n$, prove that if $v(n,h) = f_n(n,h)$, then all prime divisors $p$ of $n$ satisfy $p \equiv 1 \pmod{h}$, so $n \equiv 1 \pmod{h}$, and construct $A$ as a union of sets $A_i$ in each component of the group’s invariant factor decomposition.
• Show that $hA$ avoids the identity element by analyzing the maximal index $k$ of components contributing non-zero coordinates in any $h$-fold sum, proving $0 \notin hA$.

Experimental results

Research questions

• RQ1 What is the minimal size $m$ such that every generating $m$-subset $A$ of a finite abelian group $G$ satisfies $hA = G$?
• RQ2 How does the $h$-critical number $\chi(G,h)$ change when restricted to generating subsets, and what is the exact value of $\widehat{\chi}(G,h)$?
• RQ3 What is the minimal size $m$ such that every generating $m$-subset $A$ of $G$ satisfies $[0,s]A = G$?
• RQ4 Can the function $v(n,h)$, which bounds the size of $h$-incomplete subsets, be used to fully characterize $\widehat{\chi}(G,h)$ for all finite abelian groups?
• RQ5 Under what group-theoretic conditions (e.g., on the exponent or rank) does $\widehat{\chi}(G,h)$ differ from $\chi(G,h)$?

Key findings

• The value of $\widehat{\chi}(G,h)$ is exactly $v(n,h) + 1$, where $v(n,h) = \max_{d \mid n} \left\{ \left(\left\lfloor\frac{d-2}{h}\right\rfloor + 1\right) \cdot \frac{n}{d} \right\}$, resolving the $\widehat{\chi}(G,h)$ problem for all finite abelian groups.
• For $\widehat{\chi}(G,[0,s])$, the result is $v(n,s) + 1$, extending the known $\chi(G,s)$ result to generating subsets.
• When $n$ is prime, $\widehat{\chi}(\mathbb{Z}_n,h) = \left\lfloor\frac{n-2}{h}\right\rfloor + 2$, achieved by the set $\{1, 2, \dots, \left\lfloor\frac{n-2}{h}\right\rfloor + 1\}$, which generates $\mathbb{Z}_n$ but whose $h$-fold sumset omits $h-1$.
• For composite $n$, if $v(n,h) = f_n(n,h) = \left\lfloor\frac{n-2}{h}\right\rfloor + 1$, then $n \equiv 1 \pmod{h}$, and $\widehat{\chi}(G,h) = \frac{n-1}{h} + 1$.
• A construction of a generating $A \subset G$ of size $v(n,h)$ with $hA \neq G$ is provided via a union of sets $A_i$ in the invariant factor decomposition of $G$, ensuring $0 \notin hA$ by tracking the maximal non-zero coordinate index.
• The proof establishes that $\widehat{\chi}(G,h) = \chi(G,h)$ if and only if $v(n,h) + 1 = \chi(G,h)$, which holds when the maximal $f_d(n,h)$ is attained at $d=n$, and otherwise $\widehat{\chi}(G,h) < \chi(G,h)$.