Let us fix a prime p. The Erdős-Ginzburg-Ziv problem asks for the minimum integer s such that any collection of s points in the lattice ℤⁿ contains p points whose centroid is also a lattice point in ℤⁿ. For large n, this is essentially equivalent to asking for the maximum size of a subset of \(\mathbb{F}_p^n\) without p distinct elements summing to zero.

In this paper, we give a new upper bound for this problem for any fixed prime p ≥ 5 and large n. In particular, we prove that any subset of \(\mathbb{F}_p^n\) without p distinct elements summing to zero has size at most \({C_p} \cdot {\left( {2\sqrt p } \right)^n}\), where C_p is a constant only depending on p. For p and n going to infinity, our bound is of the form p^{(1/2)·(1+o(1))n}, whereas all previously known upper bounds were of the form p^(1−o(1))n (with pⁿ being a trivial bound).

Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds.

让我们确定一个素数p。鄂尔多斯-金兹堡-谢夫这道题的最小整数小号，使得任何集合小号晶格ℤ点^ň包含p点，其重心也ℤ格点^ñ。对于较大的n，这本质上等同于要求\（\ mathbb {F} _p ^ n \）的子集的最大大小，而没有p个不同元素的总和为零。

在本文中，对于任何固定素数p≥5和大n，我们都为该问题提供了一个新的上限。特别地，我们证明了\（\ mathbb {F} _p ^ n \）的任何子集，如果没有p个不同元素的总和为零，则其大小最多为\（{C_p} \ cdot {\ left（{2 \ sqrt p} \ right）^ n} \），其中C _p是仅取决于p的常数。对于p和n趋于无穷大，我们的边界形式为p ^{（1/2）·（1+ o（1））n}，而所有先前已知的上界形式为p ^{（1- o（1））ñ}（其中p ⁿ是一个琐碎的界线）。

我们的证明使用了所谓的多色无和定理，该定理是Croot-Lev-Pach多项式方法的结果。Naslund以及Fox和作者已经应用了这种方法及其后果，以证明本文研究的问题的范围。但是，使用一些关键的新想法，我们可以大大改善它们的范围。