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The fractional part of \(\alpha n^k\). (English) Zbl 0629.10029

It was shown by I. Danicic [Thesis, London, 1957] that if \(\varepsilon >0\) and a positive integer \(k\) are given, then for \(N\geq N(k,\varepsilon)\) and any real \(\alpha\) one can solve \(\| \alpha n^k\| \leq N^{\varepsilon -\eta}\) with \(n\leq N\) and \(\eta =2^{1-k}\). For large \(k\) one can improve the value of \(\eta\). R. C. Baker [Diophantine inequalities. Oxford: Clarendon Press (1986; Zbl 0592.10029)] has given \(\{4k(\log k+\log \log k+3)\}^{-1}\) as an admissible value.
The present paper considers the corresponding “unlocalized” results, by looking for infinitely many solutions of \(\| \alpha n^k\| \leq n^{\varepsilon -\eta}\). It is shown that one can take \(\eta =(4/3)2^{1-k}\) for \(k\geq 6\) and \(\eta =1/16k\) in general. (The latter is better when \(k\geq 9.)\) In fact for large \(k\) the constant 16 may be replaced by \(14.424\dots\) Unfortunately it does not seem possible to obtain the corresponding localized results so as to supersede the work of Danicic and Baker. The paper achieves this only for algebraic \(\alpha\).
For small \(k\) one uses the author’s recent sharpening of Weyl’s inequality [J. Lond. Math. Soc., II. Ser. 38, No. 2, 216–230 (1988; Zbl 0619.10046)]. For large \(k\) one combines ideas of I. M. Vinogradov [Izv. Akad. Nauk SSSR, Ser. Mat. 22, 161–164 (1958; Zbl 0097.26302)] used in bounding \(\sum n^{-it}\), with the method of A. A. Karatsuba [Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 5, 935–947 (1985; Zbl 0594.10041)] who produced a 1-dimensional analogue of Vinogradov’s mean value theorem for sums restricted to “well factorable” integers.

MSC:

11J71 Distribution modulo one
11L15 Weyl sums
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References:

[1] Roth, Mathematika 2 pp 1– (1955)
[2] Karatsuba, Izv. Akad. Nauk SSSR, Ser. Mat 49 pp 935– (1985)
[3] Vinogradov, The Method of Trigonometrical Sums in the Theory of Numbers (1954) · Zbl 0055.27504
[4] Baker, Diophantine Inequalities, London Math. Soc. Monographs (1986)
[5] Vinogradov, Izv. Akad. Nauk SSSR, Ser. Mat. 22 pp 161– (1958)
[6] DOI: 10.1093/qmath/os-19.1.249 · Zbl 0031.20502 · doi:10.1093/qmath/os-19.1.249
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