Davenport constant

In mathematics, the Davenport constant $D(G&hairsp;)$ is an invariant of a group studied in additive combinatorics, quantifying the size of nonunique factorizations. Given a finite abelian group $G$, $D(G&hairsp;)$ is defined as the smallest number such that every sequence of elements of that length contains a non-empty subsequence adding up to 0. In symbols, this is
 * $$D(G) = \min\left\{ N : \forall\left(\{g_n\}_{n=1}^N \in G^N\right)\left(\exists\{n_k\}_{k=1}^K : \sum_{k=1}^K{g_{n_k}} = 0\right) \right\}.$$

Example

 * The Davenport constant for the cyclic group $$G = \mathbb Z/n\mathbb Z$$ is $n$. To see this, note that the sequence of a fixed generator, repeated $n −&thinsp;1$ times, contains no subsequence with sum $0$. Thus $D(G&hairsp;) ≥ n$. On the other hand, if $$\{g_k\}_{k=1}^n$$ is an arbitrary sequence, then two of the sums in the sequence $$\left\{\sum_{k=1}^K{g_k}\right\}_{K=0}^n$$ are equal.  The difference of these two sums also gives a subsequence with sum $0$.

Properties

 * Consider a finite abelian group $G = ⊕i&thinsp;Cdi$&hairsp;, where the $d1 | d2 | ... | dr$ are invariant factors. Then
 * $$D(G) \ge M(G) = 1-r+\sum_i{d_i}.$$
 * The lower bound is proved by noting that the sequence "$d1&thinsp;−&thinsp;1$ copies of $(1, 0, ..., 0)$, $d2 −&thinsp;1$ copies of $(0, 1, ..., 0)$, etc." contains no subsequence with sum $0$.


 * $$\mathbb{z}$_{3}⊕$\mathbb{z}$_{3}⊕$\mathbb{z}$_{3d}$ for p-groups or for $D = M$.
 * $r &hairsp;=&thinsp;1,&thinsp;2$ for certain groups including all groups of the form $D = M$ and $C2&thinsp;⊕&thinsp;C2n&thinsp;⊕&thinsp;C2nm$.
 * There are infinitely many examples with $r$ at least $C3&thinsp;⊕&thinsp;C3n&thinsp;⊕&thinsp;C3nm$ where $D$ does not equal $M$; it is not known whether there are any with $4$.
 * Let $$\exp(G)$$ be the exponent of $G$. Then
 * $$\frac{D(G)}{\exp(G)} \leq 1+\log\left(\frac{|G|}{\exp(G)}\right).$$

Applications
The original motivation for studying Davenport's constant was the problem of non-unique factorization in number fields. Let $$\mathcal{O}$$ be the ring of integers in a number field, $G$ its class group. Then every element $$\alpha\in\mathcal{O}$$, which factors into at least $r = 3$ non-trivial ideals, is properly divisible by an element of $$\mathcal{O}$$. This observation implies that Davenport's constant determines by how much the lengths of different factorization of some element in $$\mathcal{O}$$ can differ.

The upper bound mentioned above plays an important role in Ahlford, Granville and Pomerance's proof of the existence of infinitely many Carmichael numbers.

Variants
Olson's constant $D(G&hairsp;)$ uses the same definition, but requires the elements of $$\{g_n\}_{n=1}^N$$ to be distinct.


 * Balandraud proved that $O(G&hairsp;)$ equals the smallest $k$ such that $$\frac{k(k+1)}{2} \geq p$$.
 * For $O(Cp&hairsp;)$ we have
 * $$O(C_p\oplus C_p) = p-1+O(C_p)$$.
 * On the other hand, if $p > 6000$ with $G = C&thinsp;r &hairsp;p$, then Olson's constant equals the Davenport constant.