Four-Variable Expanders Over The Prime Fields

Let F-p be a prime field of order p > 2, and let A be a set in F-p with very small size in terms of p. In this note, we show that the number of distinct cubic distances determined by points in A x A satisfies vertical bar(A - A)(3) + (A - A)(3 vertical bar) >> vertical bar A vertical bar(8/7), which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that max {vertical bar A + A vertical bar, vertical bar f( A, A)vertical bar} >> vertical bar A vertical bar(6/5), where f(x, y) is a quadratic polynomial in F-p[x, y] that is not of the form g(alpha x + beta y) for some univariate polynomial g.