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In this paper, we formulate and prove the so-called $p$-adic non-commutative analytic subgroup theorem. This result is seen as the $p$-adic analogue of a recent theorem given by Yafaev.

Given $E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d$, with the finite field $\mathbb{F}_q$ of order $q$ and the integer $d \ge 2$, we define the two-parameter distance set as $\Delta_{d, d}(E)=\left\{\left(\|x_1-y_1\|, \|x_2-y_2\|\right) : (x_1,x_2), (y_1,y_2) \in E \right\}$. Birklbauer and Iosevich (2017) proved that if $|E| \gg q^{\frac{3d+1}{2}}$, then $ |\Delta_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| \gg q^{\frac{10}{3}}$, then $ |\Delta_{2, 2}(E)| \gg q^2$. In this paper, we present extensions and improvements of these results.

Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and $\mathcal A$ be a set in $\mathcal R$. In this paper, we improve a recent result due to Yazici (2018) on a sum-product type problem. More precisely, we will prove that 1. If $|\mathcal A|\gg q^{r-\frac{1}{3}}$, then $$\max\left\lbrace |\mathcal A+\mathcal A|, |\mathcal A^2+\mathcal A^2|\right\rbrace \gg q^{\frac{r}{2}}|\mathcal A|^{\frac{1}{2}}.$$ 2. If $q^{r-\frac{3}{8}}\ll |\mathcal A|\ll q^{r-\frac{1}{3}}$, then $$\max\left\lbrace |\mathcal A+\mathcal A|, |\mathcal A^2+\mathcal A^2|\right\rbrace \gg \frac{|\mathcal A|^2}{q^{\frac{2r-1}{2}}}.$$ 3. If $|\mathcal A+\mathcal A||\mathcal A|^2\gg q^{3r-1}$ and $2q^{r-1}\le |\mathcal A|\ll q^{r-\frac{3}{8}}$, then $$\max\left\lbrace |\mathcal A+\mathcal A|, |\mathcal A^2+\mathcal A^2|\right\rbrace \gg q^{r/3}|\mathcal A|^{2/3}.$$

Let $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\mathcal{E}|\gg q^{d-\frac{d-1}{k+1}}$ then $\mathcal{E}$ determines a positive proportion of all $k$-simplices. In this paper, we give an improvement of this result in the case when $\mathcal{E}$ is the Cartesian product of sets. More precisely, we show that if $\mathcal{E}$ is the Cartesian product of sets and $q^{\frac{kd}{k+1-1/d}}=o(|\mathcal{E}|)$, the number of congruence classes of $k$-simplices determined by $\mathcal{E}$ is at least $(1-o(1))q^{\binom{k+1}{2}}$, and in some cases our result is sharp.

It is well-known that the W\"ustholz' analytic subgroup theorem is one of the most powerful theorems in transcendence theory. The theorem gives in a very systematic and conceptual way the transcendence of a large class of complex numbers, e.g. the transcendence of $\pi$ which is originally due to Lindemann. In this paper we revisit the $p$-adic analogue of the analytic subgroup theorem and present a proof based on the method described and developed by the authors in a recent related paper., Comment: This is a preprint of the Materials accepted for publication in "p-Adic Numbers, Ultrametric Analysis and Applications"

We use a $p$-adic analogue of the analytic subgroup theorem of W\"ustholz to deduce the transcendence and linear independence of some new classes of $p$-adic numbers. In particular we give $p$-adic analogues of results of W\"ustholz contained in [G. W\"ustholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkh\"auser Boston, Boston, MA, (1983), 329-336] and generalizations of results obtained by Bertrand in [D. Bertrand, Sous-groupes \`a un param\`etre $p$-adique de vari\'et\'es de groupe, Invent. Math. 40 (1977), no. 2, 171-193] and [D. Bertrand, Probl\`emes locaux, Soci\'et\'e Math\'ematique de France, Ast\'erisque 60-70 (1979), 163-189]., Comment: This is a preprint of the Materials accepted for publication in "Glasnik Matematicki"