On three-variable expanders over finite valuation rings

Abstract

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

Theorem 1.6 is cited because it has already proved