The Olson constant $\mathcal{𝒪}L({\mathbb{𝔽}}_p^d)$ represents the minimum positive integer $t$ with the property that every subset $A\subset {\mathbb{𝔽}}_p^d$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\mathcal{𝒪}L({\mathbb{𝔽}}_p^d)$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$.

We prove that for any fixed $d\ge 2$ and $𝜖>0$, the Olson constant of ${\mathbb{𝔽}}_p^d$ satisfies the inequality

$$\mathcal{𝒪}L({\mathbb{𝔽}}_p^d)\le (d-1+𝜖)p$$

for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

中文翻译：

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有限域上向量空间中的零和子集

奥尔森常数$\mathcal{𝒪}\mathrm{大号}({\mathbb{𝔽}}_p^d)$表示最小正整数$吨$具有每个子集的属性$\mathrm{一个}\subset {\mathbb{𝔽}}_p^d$基数的$吨$包含一个总和为零的非空子集。估计问题 $\mathcal{𝒪}\mathrm{大号}({\mathbb{𝔽}}_p^d)$是加法组合学中最古老的问题之一，即使对于这种情况也有着悠久而有趣的历史$d=1$.

我们证明对于任何固定的$d\ge 2$和$𝜖>0$，奥尔森常数${\mathbb{𝔽}}_p^d$满足不等式

$$\mathcal{𝒪}\mathrm{大号}({\mathbb{𝔽}}_p^d)\le (d-1+𝜖)p$$

对于所有足够大的素数$p$. 这解决了 Hoi Nguyen 和 Van Vu 的猜想。