标题： 改进Behrend的构造：整数和有限域中没有等差数列的集合 显示英文标题

标题： Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields

摘要： 我们证明了关于不包含三项等差数列的子集$A\subseteq \{1,\dots,N\}$或$A\subseteq \mathbb{F}_p^n$的最大大小的新下界。 在$\{1,\dots,N\}$的背景下，这是对 Behrend 于 1946 年提出的经典构造在低阶因子之外的首次改进（特别是，这是首次准多项式改进）。 在$\mathbb{F}_p^n$的设定下，对于固定的素数$p$和大的$n$，我们证明了某个绝对常数$c>1/2$的下界$(cp)^n$（对于$c = 1/2$，可以通过 1940 年代的经典构造得到这样的界限，但改进这一点是一个众所周知的开放问题）。 显示英文摘要

摘要： We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a classical construction of Behrend from 1946 beyond lower-order factors (in particular, it is the first quasipolynomial improvement). In the setting of $\mathbb{F}_p^n$ for a fixed prime $p$ and large $n$, we prove a lower bound of $(cp)^n$ for some absolute constant $c>1/2$ (for $c = 1/2$, such a bound can be obtained via classical constructions from the 1940s, but improving upon this has been a well-known open problem).