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 × A satisfies |(A - A)³ + (A - A)³| ≫ |A|^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 {|A + A|, |f(A,A)|} ≫ |A|^6/5, where f(x, y) is a quadratic polynomial in F_p[x, y] that is not of the form g(αx + βy) for some univariate polynomial g. [ABSTRACT FROM AUTHOR]