论文标题
避免$ p $ - term算术进度$ {\ mathbb z} _ {q}^n $是指数级小的
Sets avoiding $p$-term arithmetic progressions in ${\mathbb Z}_{q}^n$ are exponentially small
论文作者
论文摘要
Pach和Palincza证明了Ellenberg和Gijswijt的以下概括，并以$ k $ term算术算术的不含进度的子集约束，其中$ k \ in \ in \ {4,5,6 \} $： 令$ m> 0 $为一个整数，以至于$ 6 $划分$ m $，让$ k \ in \ {4,5,6 \} $。然后$$ r_k（{\ mathbb z} _ {m}^n）\ leq（0.948m）^n $$如果$ n $足够大。 在Pach和Palincza的上限的证明技术的基础上，我们通过以下方式概括了Ellenberg和Gijswijt的界限： 令$ p> 2 $成为任何整数，让$ q> 2 $为素数。假设$ p \ leq Q $。然后，存在$ n_0 \ in \ mathbb n $ integer和$ 0 <δ（p，q）<1 $真实数字，使得$$ r_p（{\ sathbb z} _ {q}^n）\ leq（Δ（p，p，q）q）
Pach and Palincza proved the following generalization of Ellenberg and Gijswijt's bound for the size of $k$-term arithmetic progression-free subsets, where $k\in \{4,5,6\}$: Let $m>0$ be an integer such that $6$ divides $m$ and let $k\in \{4,5,6\}$. Then $$ r_k({\mathbb Z}_{m}^n)\leq (0.948m)^n $$ if $n$ is sufficiently large. Building on the proof technique of Pach and Palincza's upper bound we generalize the Ellenberg and Gijswijt's bound in the following way: Let $p>2$ be any integer and let $q>2$ be a prime. Suppose that $p\leq q$. Then the there exists an $n_0\in \mathbb N$ integer and a $0<δ(p,q)<1$ real number such that $$ r_p({\mathbb Z}_{q}^n)\leq (δ(p,q)q)^n $$ for each $n>n_0$.
