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On three-variable expanders over finite valuation rings (2007.05251v2)
Published 10 Jul 2020 in math.CO
Abstract: Let $\mathcal{R}$ be a finite valuation ring of order $qr$. 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{ qr, \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 $$\max{ |A \cdot A|, |Ad + Ad|},\max{ |A + A|, |A2 + A2|},\max{|A-A|,|AA+AA|} \gg |A|{2/3}q{r/3},$$ and $|f(A) + A| \gg |A|{2/3}q{r/3}$ for any one variable quadratic polynomial $f$.