Capturing forms in dense subsets of finite fields

Brandon Hanson¹

¹ Department of Mathematics University of Toronto M5S 2E4 Toronto, Canada

Acta Arithmetica, Tome 160 (2013) no. 3, pp. 277-284

Résumé

An open problem of arithmetic Ramsey theory asks if given an $r$-colouring $c:\mathbb N\to \{1,\ldots ,r\}$ of the natural numbers, there exist $x,y\in \mathbb N$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More generally, one could replace $x+y$ with a binary linear form and $xy$ with a binary quadratic form. In this paper we examine the analogous problem in a finite field $\mathbb F_q$. Specifically, given a linear form $L$ and a quadratic form $Q$ in two variables, we provide estimates on the necessary size of $A\subset \mathbb F_q$ to guarantee that $L(x,y)$ and $Q(x,y)$ are elements of $A$ for some $x,y\in \mathbb F_q$.

Détail

DOI : 10.4064/aa160-3-4

Keywords: problem arithmetic ramsey theory asks given r colouring mathbb ldots natural numbers there exist mathbb apart trivial solution generally could replace binary linear form binary quadratic form paper examine analogous problem finite field mathbb specifically given linear form quadratic form variables provide estimates necessary size subset mathbb guarantee elements mathbb