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Moderate expanders over ringsJournal of Number Theory, 218
- Publisher
- de Gruyter
- Copyright
- © 2020 Walter de Gruyter GmbH, Berlin/Boston
- ISSN
- 0933-7741
- eISSN
- 1435-5337
- DOI
- 10.1515/forum-2020-0203
- Publisher site
- See Article on Publisher Site
Abstract
AbstractLet ℛ{\mathcal{R}} be a finite valuation ring of order qr{q^{r}}.In this paper, we prove that for any quadratic polynomial f(x,y,z)∈ℛ[x,y,z]{f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form axy+R(x)+S(y)+T(z){axy+R(x)+S(y)+T(z)} for some one-variable polynomials R,S,T{R,S,T}, we have|f(A,B,C)|≫min{qr,|A||B||C|q2r-1}|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any A,B,C⊂ℛ{A,B,C\subset\mathcal{R}}.We also study the sum-product type problems over finite valuation ring ℛ{\mathcal{R}}. More precisely, we show that for any A⊂ℛ{A\subset\mathcal{R}} with |A|≫qr-13{|A|\gg q^{r-\frac{1}{3}}} thenmax{|AA|,|Ad+Ad|}{\max\{|AA|,|A^{d}+A^{d}|\}}, max{|A+A|,|A2+A2|}{\max\{|A+A|,|A^{2}+A^{2}|\}}, max{|A-A|,|AA+AA|}≫|A|23qr3{\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}},and |f(A)+A|≫|A|23qr3{|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.
Journal
Forum Mathematicum – de Gruyter
Published: Jan 1, 2021