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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.

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In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that • If A is a set of M 2 (F q ) and |A| ≫ q 7/2 , then we have• If A is a set of SL 2 (F q ) and |A| ≫ q 5/2 , then we haveWe also obtain similar results for the cases of A(B + C) and A + BC, where A, B, C are sets in M 2 (F q ).

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In this paper we obtain a new lower bound on the Erdős distinct distances problem in the plane over prime fields. More precisely, we show that for any set A ⊂ F 2 p with |A| ≤ p 7/6 and p ≡ 3 mod 4, the number of distinct distances determined by pairs of points in A satisfies |∆(A)| |A| 1 2 + 149 4214 .Our result gives a new lower bound of |∆(A)| in the range |A| ≤ p 1+ 149 4065 . The main tools in our method are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in F 2 p . The latter is the new feature that allows us to improve the previous bound due