Four-variable expanders over the prime fields

Abstract

Let F p be a prime field of order p > 2, and 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 satisfieswhich 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 thatthat is not of the form g(αx + βy) for some univariate polynomial g.

Low-energy decomposition results over finite fields

“…We require the following claim, when applying Lemmas 4 and 6. A proof can be found inside the proof of [5,Theorem 1.10].…”

“…Another variation of the sum-product problem, which encodes both forms discussed above, is to consider the growth of |A+A|+|f (A, A)| for some bivariate function f . See for example [2], [5], [9] and [19] for results in this direction. Note that if f = g(ax + by) for some polynomial g, then taking A to be an arithmetic progression, one has |A + A| ≈ |f (A, A)| ≈ |A|.…”

Falconer type functions in three variables

“…For more variables polynomials, it is expectable to have bigger exponents. For instance, (x − y)(z − t) with ǫ = 1 3 + 1 13542 in [22], xy + (z − t) 2 with ǫ = 1 3 + 1 24 in [35], many other examples can also be found in [20,34].…”

A note on conditional expanders over prime fields

“…In a recent work, a very general bound for quadratic polynomials has been given by Koh, Mojarrad, Pham, and Valculescu [7].…”

“…Theorem 1.5 (Koh-Mojarrad-Pham-Valculescu, [7]). Let F p be a prime field of order p, and let f (x, y) ∈ F p [x, y] be a non-degenerate quadratic polynomial.…”