Two theorems on point-flat incidences

TL;DR

This paper advances the understanding of incidences between points and flats in real space by improving existing bounds on the number of flats spanned and incidences, contributing to combinatorial geometry.

Contribution

It provides improved bounds on point-flat incidences and the number of flats spanned by point sets, refining previous theorems by Beck and Elekes-Tóth.

Findings

Enhanced lower bounds on the number of k-flats spanned by point sets.

Improved bounds on incidences between points and k-flats.

Refined combinatorial geometric inequalities.

Abstract

We improve the theorem of Beck giving a lower bound on the number of $k$-flats spanned by a set of points in real space, and improve the bound of Elekes and T\'oth on the number of incidences between points and $k$-flats in real space.