更好的NOF精确- $N$协议的更大的无角集
Larger Corner-Free Sets from Better NOF Exactly-$N$ Protocols
摘要：
如果整数平面网格$[N] \times [N]$的一个子集不包含$(x,y), (x+δ,y), (x,y+δ)$形式的三元组，则该子集被称为无角点的。已知这样的集合具有极小的密度，但是这个密度可以有多大仍然是未知的。以前最好的构造是基于Behrend的$[N]$的大子集，没有$3$ -项等差数列。在这里，我们提供了几十年来对这个下限的第一次实质性改进。我们解决这个问题的方法是基于通信复杂性理论。在$3$ -玩家确切地- $N$问题中，玩家需要决定$x+y+z=N$是否为输入$x,y,z$和固定的$N$。这是在多人前额数字（NOF）模型中考虑的第一个问题。尽管这是一个基本问题，但多年来没有取得任何进展。直到最近才首次发现了显式协议，但迄今为止尚未实现复杂性的改善。本文提出了第一个改进的协议，完全$N$问题。这也是通信复杂性的算法思想在加法组合学中结出果实的第一个重要例子。
英文摘要：
A subset of the integer planar grid $[N] \times [N]$ is called corner-free if it contains no triple of the form $(x,y), (x+δ,y), (x,y+δ)$. It is known that such a set has a vanishingly small density, but how large this density can be remains unknown. The best previous construction was based on Behrend's large subset of $[N]$ with no $3$-term arithmetic progression. Here we provide the first substantial improvement to this lower bound in decades. Our approach to the problem is based on the theory of communication complexity. In the $3$-players exactly-$N$ problem the players need to decide whether $x+y+z=N$ for inputs $x,y,z$ and fixed $N$. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Despite the basic nature of this problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-$N$ problem. This is also the first significant example where algorithmic ideas in communication complexity bear fruit in additive combinatorics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
