Geodesic
On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders
Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 2, pp. 133-155

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In this paper we prove that for any sufficiently large $Q\in{\mathbb N}$ there exist cylinders $K\subset{\mathbb Q}_p$ with Haar measure $\mu(K)\le \frac{1}{2}Q^{-1}$ which do not contain algebraic $p$-adic numbers $\alpha$ of degree $\deg\alpha=n$ and height $H(\alpha)\le Q$. The main result establishes in any cylinder $K$, $\mu(K)>c_1Q^{-1}$, $c_1>c_0(n)$, the existence of at least $c_{3}Q^{n+1}\mu(K)$ algebraic $p$-adic numbers $\alpha\in K$ of degree $n$ and $H(\alpha)\le Q$. 
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     author = {N. V. Budarina and F. G\"otze},
     title = {On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
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     number = {2},
     year = {2015},
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N. V. Budarina; F. Götze. On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders. Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 2, pp. 133-155. http://geodesic.mathdoc.fr/item/DVMG_2015_15_2_a0/