Three-variable expanding polynomials and higher-dimensional distinct distances
We determine which quadratic polynomials in three variables are expanders over an
arbitrary field F F. More precisely, we prove that for a quadratic polynomial f∈ FF x, y, z,
which is not of the form g (h (x)+ k (y)+ l (z)), we have| f (A× B× C)|≫ N 3/2 for any sets A, B,
C⊂ FF 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+ A 2| and
max| A+ A|,| A 2+ A 2|, and we prove that a Cartesian product A×...× A⊂ F^ d F d determines …
arbitrary field F F. More precisely, we prove that for a quadratic polynomial f∈ FF x, y, z,
which is not of the form g (h (x)+ k (y)+ l (z)), we have| f (A× B× C)|≫ N 3/2 for any sets A, B,
C⊂ FF 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+ A 2| and
max| A+ A|,| A 2+ A 2|, and we prove that a Cartesian product A×...× A⊂ F^ d F d determines …
Abstract
We determine which quadratic polynomials in three variables are expanders over an arbitrary field . More precisely, we prove that for a quadratic polynomial 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 ⊂ 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 ⊂ determines almost |A|2 distinct distances if |A| is not too large.
Springer
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