Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances

Abstract

We determine which quadratic polynomials in three variables are expanders over an arbitrary field \(\mathbb{F}\). More precisely, we prove that for a quadratic polynomial f\(\mathbb{F}\)[x,y,z], which is not of the form g(h(x)+k(y)+l(z)), we have |f(A×B×C)|≫N3/2 for any sets A,B,C\(\mathbb{F}\) with |A|=|B|=|C|=N, with N not too large compared to the characteristic of F.

We give several related proofs involving similar ideas. We obtain new lower bounds on |A+A2| and max{|A+A|, |A2+A2|}, and we prove that a Cartesian product A×...×A\(\mathbb{F}^d\) determines almost |A|2 distinct distances if |A| is not too large.

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Correspondence to Thang Pham.

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Pham, T., Vinh, L.A. & de Zeeuw, F. Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances. Combinatorica 39, 411–426 (2019). https://doi.org/10.1007/s00493-017-3773-y

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Mathematics Subject Classification (2010)

  • 52C10