标题： 结构化的塞梅雷迪-特洛特对于格及其推广 显示英文标题

标题： Structural Szemerédi-Trotter for Lattices and their Generalizations

摘要： 当点集是整数网格的一个截面时，我们完全刻画了具有$\Theta(n^{4/3})$束发的点线构型。 这可以看作是结构化 Szemerédi-Trotter 问题的主要特例。 我们也对若干推广得到了部分刻画：(i) 当点集是算术级数与任意集合的笛卡尔积时，排除了共点线的情况。(ii) 我们研究了其中一个或两个集合是广义算术级数的笛卡尔积的情形。 我们的证明依赖于推导乘法能量的性质。 显示英文摘要

摘要： We completely characterize point--line configurations with $\Theta(n^{4/3})$ incidences when the point set is a section of the integer lattice. This can be seen as the main special case of the structural Szemer\'edi-Trotter problem. We also derive a partial characterization for several generalizations: (i) We rule out the concurrent lines case when the point set is a Cartesian product of an arithmetic progression and an arbitrary set. (ii) We study the case of a Cartesian product where one or both sets are generalized arithmetic progression. Our proofs rely on deriving properties of multiplicative energies.