Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances

  • Thang Pham
  • Le Anh Vinh
  • Frank de Zeeuw


We determine which quadratic polynomials in three variables are expanders over an arbitrary field F. More precisely, we prove that for a quadratic polynomial f∈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⊂ 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⊂F d determines almost |A|2 distinct distances if |A| is not too large.

Mathematics Subject Classification (2010)



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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics cole polytechnique fdrale de LausanneLausanneSwitzerland
  2. 2.Vietnam National Institute of Educational SciencesHanoiVietnam

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