Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances

Abstract

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)| ≫ N 3/2 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 applications. We use this result for f = (x − y) 2 + z to obtain new lower bounds on |A + A 2 | and max{|A + A|, |A 2 + A 2 |}, and to prove that a Cartesian product A × · · · × A ⊂ F d determines almost |A| 2 distinct distances if |A| is not too large.

Expanding phenomena over matrix rings

“…Note that all the M j and M 2i that appear in (16) are adjacency matrices of regular graphs. Thus all of them share an eigenvector v 1 = (1, .…”

On the determinants and permanents of matrices with restricted entries over prime fields

“…d 2d−1 , is indicated to be true for the Erdős distinct distances problem in A d over F d q (see [4]). Recently, Pham, Vinh, and De Zeeuw [10] showed that for A ⊂ F p , the number of distinct distances determined by points in A d is almost |A| 2 if the size of A is not so large. Thus it seems reasonable to make the following conjecture.…”

Four-variable expanders over the prime fields

“…In [12], Pham, Vinh and de Zeeuw obtained a more general result. More precisely, they showed that for A, B, C ⊂ F p with |A| = |B| = |C| = N ≤ p 2/3 , and for any quadratic polynomial in three variables f (x, y, z) ∈ F p [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 .…”