Capturing forms in dense subsets of finite fields

• Autorzy: Brandon Hanson
• Języki publikacji: EN

Abstrakty

EN

An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → {1,...,r} of the natural numbers, there exist x,y ∈ 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 $𝔽_q$. Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size of $A ⊂ 𝔽_q$ to guarantee that L(x,y) and Q(x,y) are elements of A for some $x,y ∈ 𝔽_q$.

Kategorie tematyczne

• 05D10: Ramsey theory
• 11B30: Arithmetic combinatorics; higher degree uniformity

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 160
• Numer: 3
• Strony: 277-284

Daty

wydano

2013

Twórcy

autor

Brandon Hanson

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

Identyfikatory

DOI

10.4064/aa160-3-4