For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq \mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such a set is at most $\frac{C}{(\log\log\log n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/\log^{*} n)$, which follows from the density Hales-Jewett theorem.

翻译：对于素数 $p$，$\mathbb{F}_p^n$ 中的一个受限算术数列是指向量三元组 $x, x+a, x+2a$，其中公差 $a$ 是取自 $\{0,1,2\}^n$ 的非零元素。一个不含受限算术数列的最大子集 $A\subseteq \mathbb{F}_p^n$ 的规模是多少？我们证明，任何此类集合的密度至多为 $\frac{C}{(\log\log\log n)^c}$，其中 $c,C>0$ 仅依赖于 $p$，从而为这类集合的密度给出了首个合理界。此前，已知的最佳界为 $O(1/\log^{*} n)$，该结果源于密度 Hales-Jewett 定理。