论文标题
指数较大的仿射和投影帽
Exponentially Larger Affine and Projective Caps
论文作者
论文摘要
尽管最近在CAP套件的上限上取得了突破（Crot，Lev和Pach（2017）以及Ellenberg和Gijswijt（2017）），经典的CAP SET构造并未受到影响。在这项工作中，我们引入了一种与奇数模量$ p $的所有仿射空间中的帽子的截然不同的构造方法。此外，我们表明，对于所有Primes $ p \ equiv 5 \ bmod 6 $，带有$ p \ leq 41 $，新构造导致仿期和投射帽的增长额定成倍更大，在$ \ mathrm {agrm {ag}（n，n，p）$和$ \ mathrm {pg}（pg}（n，n，p）（n，p）$中。例如，当$ p = 23 $时，具有增长$（8.0875 \ ldots）^n $的上限是Bose（1947）的三维示例，唯一的改进是基于六维的示例，而Edel（2004）的$（8.0901 \ ldots）^n $。我们将其提高到$（9-O（1））^n $。
In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach (2017) and Ellenberg and Gijswijt (2017)), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus $p$. Moreover, we show that for all primes $p \equiv 5 \bmod 6$ with $p \leq 41$, the new construction leads to an exponentially larger growth of the affine and projective caps in $\mathrm{AG}(n,p)$ and $\mathrm{PG}(n,p)$. For example, when $p=23$, the existence of caps with growth $(8.0875\ldots)^n$ follows from a three-dimensional example of Bose (1947), and the only improvement had been to $(8.0901\ldots)^n$ by Edel (2004), based on a six-dimensional example. We improve this lower bound to $(9-o(1))^n$.
