Semialgebraic Methods and Generalized Sum-Product Phenomena

Abstract

For a bivariate [Formula Presented] our first result shows that for all finite A ⊆ R, |P(A,A)| ≥ α|A|5/4 with α = α(degP) ∈ R>0 unless P(x,y) = f (γu(x)+δu(y)) or P(x,y) = f (um(x)un(y)) for some univariate f,u ∈ R[t] \R, constants γ,δ ∈R≠0, and m,n ∈N≥1. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results are sum-product type theorems for two polynomials, generalizing the classical result by Erdős and Szemerédi as well as a theorem by Shen. We also obtain similar results for C, and from this deduce results for fields of characteristic 0 and fields of large prime characteristic. The proofs of our results use tools from semialgebraic/o-minimal geometry