Inverse zero-sum problems II

Let $G$ be an additive finite abelian group. A sequence over $G$ is called a minimal zero-sum sequence if the sum of its terms is zero and no proper subsequence has this property. Davenport's constant of $G$ is the maximum of the lengths of the minimal zero-sum sequences over $G$. Its value is well-known for groups of rank two. We investigate the structure of minimal zero-sum sequences of maximal length for groups of rank two. Assuming a well-supported conjecture on this problem for groups of the form $C_m \oplus C_m$, we determine the structure of these sequences for groups of rank two. Combining our result and partial results on this conjecture, yields unconditional results for certain groups of rank two.Comment: new version contains results related to Davenport's constant only; other results will be described separatel

Some remarks on barycentric-sum problems over cyclic groups

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g_1, ..., g_k} satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.Comment: to appear in European Journal of Combinatoric

Zero-sum problems with congruence conditions

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum subsequence $T$ of length $|T| \equiv 0 \mod d$. We determine $\mathsf s_{d \mathbb N} (G)$ for all $d\geq 1$ when $G$ has rank at most two and, under mild conditions on $d$, also obtain precise values in the case of $p$-groups. In the same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv constant provided that, for the $p$-subgroups $G_p$ of $G$, the Davenport constant $\mathsf D (G_p)$ is bounded above by $2 \exp (G_p)-1$. This generalizes former results for groups of rank two

On the Olson and the Strong Davenport constants

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant

Circuit theory of crossed Andreev reflection

We consider transport in a three terminal device attached to one superconducting and two normal metal terminals, using the circuit theory of mesoscopic superconductivity. We compute the nonlocal conductance of the current out of the first normal metal terminal in response to a bias voltage between the second normal metal terminal and the superconducting terminal. The nonlocal conductance is given by competing contributions from crossed Andreev reflection and electron cotunneling, and we determine the contribution from each process. The nonlocal conductance vanishes when there is no resistance between the superconducting terminal and the device, in agreement with previous theoretical work. Electron cotunneling dominates when there is a finite resistance between the device and the superconducting reservoir. Decoherence is taken into account, and the characteristic timescale is the particle dwell time. This gives rise to an effective Thouless energy. Both the conductance due to crossed Andreev reflection and electron cotunneling depend strongly on the Thouless energy. We suggest to experimentally determine independently the conductance due to crossed Andreev reflection and electron cotunneling in measurement of both energy and charge flow into one normal metal terminal in response to a bias voltage between the other normal metal terminal and the superconductor.Comment: v2: Published version with minor changes, 12 pages and 9 figure

Remarks on a generalization of the Davenport constant

A generalization of the Davenport constant is investigated. For a finite abelian group $G$ and a positive integer $k$, let $D_k(G)$ denote the smallest $\ell$ such that each sequence over $G$ of length at least $\ell$ has $k$ disjoint non-empty zero-sum subsequences. For general $G$, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence $(D_k(G))_{k\in\mathbb{N}}$ is eventually an arithmetic progression with difference $\exp(G)$, and several questions arising from this fact are investigated. For elementary 2-groups, $D_k(G)$ is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).Comment: Various expository changes, updated and slightly expanded bibliograph