Unions of lines in F^n

TL;DR

This paper establishes sharp bounds on the dimension of unions of lines in finite field vector spaces, extending to k-planes, with implications for understanding geometric configurations over finite fields.

Contribution

It provides the first sharp estimate for the dimension of unions of lines in finite fields without structural assumptions, and extends results to unions of k-planes.

Findings

Dimension of union of lines is at least d + beta under given conditions

Sharp estimate achieved for unions of lines in finite fields

Results extended to unions of k-planes

Abstract

We show that if a collection of lines in a vector space over a finite field has "dimension" at least 2(d-1) + beta, then its union has "dimension" at least d + beta. This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of k-planes.