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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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