The exact value of the Davenport constant of homogeneous groups: the conjecture $\mathsf{D}(C_n^r) = r(n-1)+1$

The Davenport constant $\mathsf{D}(G)$ of a finite abelian group $G$ is the least length that forces every sequence over $G$ to contain a non-empty zero-sum subsequence. For the homogeneous groups $C_n^r$ (direct sums of $r$ cyclic groups of order $n$), the classical conjecture $\mathsf{D}(C_n^r) = r(n-1)+1$, open since the late 1960s, is confirmed only when $n$ is a prime power, when $r \le 2$, and at rank 3 when $n = 2p^\alpha$ or $n = 2^\alpha\cdot 3$. The source paper (Girard, 2018) proved the asymptotic $\mathsf{D}(C_n^r) \sim rn$ for fixed $r$, with an error term controlled by the Alon–Dubiner constants $c_r$: $\mathsf{D}(C_n^r) \le r(n-1)+1 + (c_r - r)(n/\mathcal{P}(n)-1)$, where $\mathcal{P}(n)$ is the largest prime-power divisor of $n$. The conjecture remains open; the Alon–Dubiner, Han–Zhang and Chao Liu results bound it without settling it. The companion questions on exact $\eta(C_n^r)$ values and on the conjectured bound $c_r \le d^r$ are likewise open.

Background

Let $C_n$ denote the cyclic group $\mathbb{Z}/n\mathbb{Z}$ and let $C_n^r = C_n \oplus \cdots \oplus C_n$ be the direct sum of $r$ copies of it. A sequence over a finite abelian group $G$ is a finite string of elements of $G$ with repetitions allowed, and a zero-sum subsequence is a non-empty selection of its terms whose sum is the identity element. The Davenport constant $\mathsf{D}(G)$ is the smallest $t$ such that every sequence over $G$ of length at least $t$ contains a non-empty zero-sum subsequence; equivalently, it is one-plus the maximal length of a zero-sumfree sequence. It is a central invariant of zero-sum theory in additive combinatorics and has a direct arithmetic meaning: for the ring of integers of a number field with ideal class group $G$, $\mathsf{D}(G)$ is the maximum number of prime ideal factors of an irreducible element, so it governs the statistics of non-unique algebraic factorizations.

Every finite abelian group decomposes as $G \cong C_{n_1} \oplus \cdots \oplus C_{n_r}$ with $1 < n_1 \mid \cdots \mid n_r$; $r$ is the rank and $n_r$ the exponent of $G$. Taking $n_i - 1$ copies of the $i$-th basis element gives a zero-sumfree sequence, so $\mathsf{D}(G) \ge \mathsf{D}^*(G) := 1 + \sum_i (n_i - 1)$; the classical general upper bound is $\mathsf{D}(G) \le n_r(1 + \log(|G|/n_r))$. Equality $\mathsf{D}(G) = \mathsf{D}^*(G)$ holds when $G$ is a $p$-group (a group whose order is a power of a prime $p$; Olson 1969) and when $G$ has rank at most 2 (Olson 1969), and it fails for infinitely many groups of higher rank with mixed invariant factors — but no group of rank 3 and no group of homogeneous form $C_n^r$ is known where it fails. Since the late 1960s (first raised in 1969 technical reports of the Mathematisch Centrum Amsterdam, and stated as Conjecture 3.5 of the Gao–Geroldinger 2006 survey; see the discussion and references in Girard 2018), it has been conjectured that equality always holds for homogeneous groups:

$\mathsf{D}(C_n^r) = r(n-1) + 1.$

This is known for all $r$ when $n$ is a prime power, for all $n$ when $r \le 2$, and for $r = 3$ when $n = 2p^\alpha$ ($p$ prime, $\alpha \ge 1$) or $n = 2^\alpha \cdot 3$ ($\alpha \ge 2$); it is open for every other pair with composite $n$, already for $\mathsf{D}(C_6^4)$ and $\mathsf{D}(C_{15}^3)$.

Two companion invariants drive the only known approach. The short zero-sum constant $\eta(G)$ is the smallest $t$ such that every sequence of length at least $t$ has a non-empty zero-sum subsequence of length at most $\exp(G)$; the inductive method bounds $\mathsf{D}(G) \le \exp(G/H)(\mathsf{D}(H)-1) + \eta(G/H)$ for a subgroup $H$. Alon and Dubiner (1995) proved $\eta(C_n^r) \le c_r(n-1)+1$ for constants $c_r$ with $c_r \le (c\, r \log r)^r$ for an absolute $c$, and conjectured that $c_r \le d^r$ for an absolute $d$. Girard (2018) combined these facts: with $\mathcal{P}(n)$ the largest prime-power divisor of $n$, $\mathsf{D}(C_n^r) \le r(n-1)+1 + (c_r - r)(n/\mathcal{P}(n) - 1)$, and therefore $\mathsf{D}(C_n^r) \sim rn$ as $n \to \infty$ for fixed $r$ — the source paper's main theorem. A cautionary fact for the companion problem: the naive formula $\eta(C_n^r) = (2^r-1)(n-1)+1$ is false in general — for $n = 3$ the values are tied to the cap-set problem (maximal sets in $\mathbb{F}_3^r$ with no three collinear points), and already $\eta(C_3^4) = 39 > 31$ — so exact $\eta$-values are a genuine determination problem rather than a two-valued conjecture.

Problem Statement

Determine the exact value of the Davenport constant $\mathsf{D}(C_n^r)$ for every pair of integers $n \ge 2$, $r \ge 2$. Here $C_n = \mathbb{Z}/n\mathbb{Z}$ is the cyclic group of order $n$, $C_n^r = C_n \oplus \cdots \oplus C_n$ is the direct sum of $r$ copies of it, and $\mathsf{D}(G)$ denotes the smallest integer $t$ such that every sequence of at least $t$ elements of $G$ (a finite string of elements, repetitions allowed) contains a non-empty subsequence whose sum is the identity element; a sequence with no such subsequence is called zero-sumfree. Equivalently: prove or refute the classical conjecture for homogeneous groups (Conjecture 1 of Girard 2018, previously Conjecture 3.5 of the Gao–Geroldinger 2006 survey; Girard states it for all integers $n, r \ge 1$, the boundary cases being trivial since $\mathsf{D}(C_n) = n$ for the cyclic group):

$\mathsf{D}(C_n^r) = r(n-1) + 1 \quad \text{for all integers } n \ge 2,\ r \ge 2.$

The lower bound $r(n-1)+1$ is classical (take $n-1$ copies of each element of a basis of $C_n^r$), so the decisive content is the upper bound: every sequence of $r(n-1)+1$ elements of $C_n^r$ contains a non-empty zero-sum subsequence. The conjecture is known for prime-power $n$ (all $r$), for all $n$ when $r \le 2$, and at rank 3 when $n = 2p^\alpha$ ($p$ prime, $\alpha \ge 1$) or $n = 2^\alpha \cdot 3$ ($\alpha \ge 2$); it is open for every other pair with composite $n$, already for $\mathsf{D}(C_6^4)$ (conjectured value 21) and $\mathsf{D}(C_{15}^3)$ (conjectured value 43). A resolution is either (a) a proof of the equality for all pairs, or (b) an explicit pair $(n, r)$ together with an explicit zero-sumfree sequence of length at least $r(n-1)+1$ over $C_n^r$, its elements given as vectors in $\{0, \dots, n-1\}^r$, which refutes the conjecture. The two companion open problems on the same groups — the exact values of the short zero-sum constant $\eta(C_n^r)$ in intermediate regimes and the Alon–Dubiner bound $c_r \le d^r$ — are not part of the target.

References

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