Three-variable expanding polynomials and higher-dimensional distinct distances

الأصل · EN

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 applications. We use this result for f=(x-y)²+z to obtain new lower bounds on |A+A²| and {|A+A|,|A²+A²|}, and to prove that a Cartesian product A× × A⊂ Fᵈ determines almost |A|² distinct distances if |A| is not too large.