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On the distribution of \(\alpha p^ k\) modulo 1. (English) Zbl 0880.11052

Let \(\alpha\) be an irrational real number and \(\beta\) be any real number. For any real \(x\) write \(|x|= \min_{n\in \mathbb{Z}} |x-n |\). Using a sieve technique developed by G. Harman [J. Lond. Math. Soc., II. Ser. 27, 9-18 (1983; Zbl 0504.10018)], the author proves that for any integer \(k\geq 3\) and any \(\varepsilon>0\) there are infinitely many primes \(p\) such that \[ |\alpha p^k+ \beta|<p^{- \sigma+ \varepsilon} \] where \[ \sigma = \begin{cases} 5/56 \quad & \text{if } k=3, \\ 1/21 \quad & \text{if } k=4, \\ (0.815)2^{-k} \quad & \text{if } k\geq 5. \end{cases} \] For \(3\leq k\leq 12\), the above result is an improvement of a result by R. C. Baker and G. Harman [Mathematika 38, 170-184 (1991; Zbl 0751.11037)] who obtained \(\sigma= (2/3)2^{-k}\).

MSC:

11J71 Distribution modulo one
11J54 Small fractional parts of polynomials and generalizations
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References:

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[2] DOI: 10.1006/jnth.1995.1020 · Zbl 0822.11062
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[8] DOI: 10.1112/plms/s3-72.2.241 · Zbl 0874.11052
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[14] DOI: 10.1006/jnth.1993.1075 · Zbl 0786.11042
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