For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S=g₁·...·g_l$ over G such that $(X^{g₁}-a₁)·...·(X^{g_l}-a_l)\ne 0\in K\left[G\right]$ for all $a₁,...,a_l\in K^{\times }$. If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for which (G) -...

We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between these dimensions by improving an inequality of Lev and Yuster, and we show that these bounds are asymptotically sharp, using in particular the existence of large dissociated subsets of {0,1}ⁿ ⊂ ℤⁿ.

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...

We show that for any relatively prime integers 1 ≤ p < q and for any finite A ⊂ ℤ one has ${|p·A+q·A|\ge (p+q)\left|A\right|-\left(pq\right)}^{(p+q-3)(p+q)+1}$.