Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sum-Product Type Estimates for Subsets of Finite Valuation Rings   Let $R$ be a finite valuation ring of order $q^r.$ Using a point-plane
incidence estimate in $R^3$, we obtain sum-product type estimates for subsets
of $R$. In particular, we prove that for $A\subset R$, $$|AA+A|\gg
\min\left\{q^{r}, \frac{|A|^3}{q^{2r-1}}\right\}.$$ We also show that if
$|A+A||A|^{2}>q^{3r-1}$, then $$|A^2+A^2||A+A|\gg
q^{\frac{r}{2}}|A|^{\frac{3}{2}}.$$

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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess
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The categories of this paper are: cs
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol