Growth Estimates in Positive Characteristic via Collisions

An application of the sum-product phenomenon to sets having no solutions of several linear equations

“…If we consider the case of prime p, then a multiplicative subgroup Γ ⊆ F * p without solutions of the equation x+y = z was constructed with the constraint |Γ| ≫ p 1/3 only, see [2]. Conjecturally, the right bound here is |Γ| p 1/2−o (1) or even p 2/3−o (1) , see [2,Section 4].…”

“…Now we need in some sum-product results. In [1] it was shown that the main theorem from [9] implies the following weaker version of the Szemerédi-Trotter Theorem for a general field F.…”

“…. , x k ) ∈ A k satisfying (1). Sets avoiding linear equations is a well-known subject of Additive Combinatorics and Number Theory, see, e.g., classical papers [11,12] about this question.…”

“…Sets avoiding linear equations is a well-known subject of Additive Combinatorics and Number Theory, see, e.g., classical papers [11,12] about this question. It is known that if b = 0 and c 1 + • • • + c k = 0, then |A| = o(p) as p → ∞ but in the other cases one can easily construct a set of positive density avoiding (1). In this paper we have to deal with the case k = 3 but instead of one equation we consider several, say, t of them.…”

“…The method of the proof is based on precise incidences results from [9] and some applications of these results from [1], [7], [10]. Usually, theorems of such a sort have to deal with small subsets of F p .…”