Davenport constant

In mathematics, the Davenport constant D(G ) is an invariant of a group studied in additive combinatorics, quantifying the size of nonunique factorizations. Given a finite abelian group G, D(G ) 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^[1]

${\displaystyle D(G)=\min \left\{N:\forall \left(\{g_{n}\}_{n=1}^{N}\in G^{N}\right)\left(\exists \{n_{k}\}_{k=1}^{K}:\sum _{k=1}^{K}{g_{n_{k}}}=0\right)\right\}.}$

Example

• The Davenport constant for the cyclic group ${\displaystyle G=\mathbb {Z} /n\mathbb {Z} }$ is n. To see this, note that the sequence of a fixed generator, repeated n − 1 times, contains no subsequence with sum 0. Thus D(G ) ≥ n. On the other hand, if ${\displaystyle \{g_{k}\}_{k=1}^{n}}$ is an arbitrary sequence, then two of the sums in the sequence ${\displaystyle \left\{\sum _{k=1}^{K}{g_{k}}\right\}_{K=0}^{n}}$ are equal. The difference of these two sums also gives a subsequence with sum 0.^[2]

Properties

• Consider a finite abelian group G = ⊕_i C_di , where the d₁ | d₂ | ... | d_r are invariant factors. Then

${\displaystyle D(G)\geq M(G)=1-r+\sum _{i}{d_{i}}.}$

The lower bound is proved by noting that the sequence "d₁ − 1 copies of (1, 0, ..., 0), d₂ − 1 copies of (0, 1, ..., 0), etc." contains no subsequence with sum 0.^[3]

• D = M for p-groups or for r  = 1, 2.
• D = M for certain groups including all groups of the form C₂ ⊕ C_2n ⊕ C_2nm and C₃ ⊕ C_3n ⊕ C_3nm.
• There are infinitely many examples with r at least 4 where D does not equal M; it is not known whether there are any with r = 3.^[3]
• Let ${\displaystyle \exp(G)}$ be the exponent of G. Then^[4]

${\displaystyle {\frac {D(G)}{\exp(G)}}\leq 1+\log \left({\frac {|G|}{\exp(G)}}\right).}$

Applications

The original motivation for studying Davenport's constant was the problem of non-unique factorization in number fields. Let ${\displaystyle {\mathcal {O}}}$ be the ring of integers in a number field, G its class group. Then every element ${\displaystyle \alpha \in {\mathcal {O}}}$, which factors into at least D(G ) non-trivial ideals, is properly divisible by an element of ${\displaystyle {\mathcal {O}}}$. This observation implies that Davenport's constant determines by how much the lengths of different factorization of some element in ${\displaystyle {\mathcal {O}}}$ can differ.^{[5][citation needed]}

The upper bound mentioned above plays an important role in Ahlford, Granville and Pomerance's proof of the existence of infinitely many Carmichael numbers.^[4]

Variants

Olson's constant O(G ) uses the same definition, but requires the elements of ${\displaystyle \{g_{n}\}_{n=1}^{N}}$ to be distinct.^[6]

• Balandraud proved that O(C_p ) equals the smallest k such that ${\displaystyle {\frac {k(k+1)}{2}}\geq p}$.
• For p > 6000 we have

${\displaystyle O(C_{p}\oplus C_{p})=p-1+O(C_{p})}$.

On the other hand, if G = C^r
_p with r ≥ p, then Olson's constant equals the Davenport constant.^[7]