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Capturing forms in dense subsets of finite fields

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Hanson B.

Acta Arithmetica

2013

tom 160

nr 3

277-284

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 $x,y ∈ 𝔽_q$.

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1 Acta Arithmetica

1 Hanson B.

1 2013

Wyniki wyszukiwania

Capturing forms in dense subsets of finite fields