abstract:

We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of Elekes, Borenstein and Croot, and Amirkhanyan, Bush, Croot, and Pryby.

The upper bounds are based on a version of the asymmetric Balog-Szemer\'{e}di-Gowers theorem for {\it group actions} combined with product theorems for the affine group. The lower bounds are based on a connection between rich lines in Cartesian product sets and {\it amenability} (or expanding families of graphs in the finite field case).

As an application of our upper bounds for rich lines in grids, we give a geometric proof of the asymmetric sum-product estimates of Bourgain and Shkredov.

摘要：

我们证明了在笛卡尔乘积点集中丰富的一般位置的行数的上限和下限。这反驳了Solymosi的猜想，并改善了Elekes，Borenstein和Croot以及Amirkhanyan，Bush，Croot和Pryby的工作。

上限基于{\ it group actions}的非对称Balog-Szemer \'{e} di-Gowers定理的一个版本以及仿射组的乘积定理。下限基于笛卡尔乘积集中的丰富线与{\ it适应性}（或在有限域情况下扩展图族）之间的联系。

作为网格上丰富线的上限的应用，我们给出了布尔加因和什克雷多夫的不对称求和估计的几何证明。