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Improved bounds for progression-free sets in $$C_8^{n}$$C8n
Israel Journal of Mathematics ( IF 0.8 ) Pub Date : 2020-02-12 , DOI: 10.1007/s11856-020-1977-0
Fedor Petrov , Cosmin Pohoata

Let $G$ be a finite group, and let $r_{3}(G)$ represent the size of the largest subset of $G$ without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$, where $C_{m}$ denotes the cyclic group of order $m$. For finite abelian groups $G \cong \prod_{i=1}^{n} C_{m_{i}}$, where $m_{1},\ldots,m_{n}$ denote positive integers such that $m_{1} | \ldots | m_{n}$, this also yields a bound of the form $r_{3}(G) \leqslant (0.903)^{\operatorname{rk}_{4}(G)} |G|$, with $\operatorname{rk}_{4}(G)$ representing the number of indices $i \in \left\{1,\ldots,n\right\}$ with $4\ |\ m_{i}$. In particular, $r_{3}(C_{8}^{n}) \leqslant (7.22)^{n}$. In this paper, we provide an exponential improvement for this bound, namely $r_{3}(C_{8}^{n}) \leq (7.09)^{n}$.

中文翻译:

改进了 $$C_8^{n}$$C8n 中无进展集的边界

令 $G$ 是一个有限群,并令 $r_{3}(G)$ 代表没有非平凡三项级数的 $G$ 的最大子集的大小。在最近的突破中,Croot、Lev 和 Pach 证明了 $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$,其中 $C_{m}$ 表示循环群订购 $m$。对于有限阿贝尔群 $G \cong \prod_{i=1}^{n} C_{m_{i}}$,其中 $m_{1},\ldots,m_{n}$ 表示正整数,使得 $m_ {1} | \ldots | m_{n}$,这也产生 $r_{3}(G) \leqslant (0.903)^{\operatorname{rk}_{4}(G)} |G|$ 形式的边界,其中 $\ operatorname{rk}_{4}(G)$ 表示索引数 $i \in \left\{1,\ldots,n\right\}$ 和 $4\ |\ m_{i}$。特别是 $r_{3}(C_{8}^{n}) \leqslant (7.22)^{n}$。在本文中,我们为这个界限提供了指数改进,即 $r_{3}(C_{8}^{n}) \leq (7.09)^{n}$。
更新日期:2020-02-12
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