On three-variable expanders over finite valuation rings

Let R be a finite valuation ring of order q(r). In this paper, we prove that for any quadratic polynomial f(x, y, z) epsilon R [x, y, z] that is of the form axy + R(x) + S(y) + T(z) for some one-variable polynomials R, S, T, we have, vertical bar f(A, B, C)vertical bar >> min {q(r), vertical bar A parallel to B parallel to C vertical bar/q(2r-1)} for any A, B, C subset of R. We also study the sum-product type problems over finite valuation ring R. More precisely, we show that for any A subset of R with vertical bar A vertical bar >> q(r-1/3) then max{vertical bar AA vertical bar, vertical bar A(d) + A(d)vertical bar}, max{vertical bar A + A vertical bar, vertical bar A(2) + A(2)vertical bar}, max{vertical bar A - A vertical bar, vertical bar AA + AA vertical bar} >> vertical bar A vertical bar(2/3) q(r/3), and vertical bar f(A) + A vertical bar >> vertical bar A vertical bar(2/3) q(r/3) for any one variable quadratic polynomial f.