A polynomial method approach to zero-sum subsets in $\mathbb {F}_{p}^{2}$
Volume 182 / 2018
Acta Arithmetica 182 (2018), 243-247
MSC: Primary 11P70; Secondary 05D99.
DOI: 10.4064/aa170309-5-12
Published online: 22 January 2018
Abstract
We prove that every subset of $\mathbb{F}_p^2$ having a nonempty intersection with each of the $p+1$ lines passing through the origin has a zero-sum subset. This is motivated by a result of Gao, Ruzsa and Thangadurai which states that $\mathit{OL}(\mathbb{F}_{p}^{2})=p+\mathit{OL}(\mathbb{F}_{p})-1$ for sufficiently large primes $p$. Here $\mathit{OL}(G)$ denotes the so-called Olson constant of the additive group $G$ and represents the smallest integer such that no subset of cardinality $\mathit{OL}(G)$ is zero-sum-free. Our proof is in the spirit of the Combinatorial Nullstellensatz.
