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Diophantine inequalities in imaginary quadratic number fields. (English) Zbl 0822.11050
The paper is closely related to the paper of E. Hlawka [Österreich. Akad. Wiss., Math.-Naturw. Kl., S.-Ber. II a 156, 255-262 (1948; Zbl 0036.308)] and generalizes a result from the quoted paper concerning the approximation of elements from $$\mathbb{Q}(i)$$ by square roots of Gaussian integers to approximation by $$n$$-th roots of these integers.
Let $$\eta\in \mathbb{Q} (i\sqrt {d})$$, $$n\geq 2$$. The author shows that there exists a constant $$C= C(n,d, \eta)>0$$ such that either $\eta- \root n\of {z}\in \mathbb{Z}(i \sqrt{d}) \quad \text{or} \quad \|\eta- \root n\of {z}\|> {C\over {| z|^{(n-1)/n}}},$ where $$\root n\of {z}$$ is the principal value of $$z$$, $$\mathbb{Z}(i \sqrt {d})$$ denotes the ring of integers in $$\mathbb{Q}(i \sqrt {d})$$, and $$\| v\|= \min\{ | v- z|$$: $$z\in \mathbb{Z} (i \sqrt{d})\}$$.
##### MSC:
 11J17 Approximation by numbers from a fixed field 11J25 Diophantine inequalities
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##### References:
 [1] HLAWKA E.: Über Folgen von Quadratwurzeln komplexer Zahlen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II Vol. 155 (1947), 255-262. · Zbl 0036.30802 [2] KOKSMA J. F.: Über die asymptotische Verteilung gewisser Zahlenfolgen modulo Eins. Nieuw Arch. Wisk. (4) 20 (1940), 179-183. · Zbl 0022.30902 [3] KOKSMA J. F.: Diophantische Approximationen. Springer, Berlin, 1936. · Zbl 0012.39602 [4] MAHLER K.: Note on the sequence $$\sqrt{n} (mod 1)$$. Nieuw Arch. Wisk. (4) 20 (1940), 176-178. · Zbl 0022.30901
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