Wong, K. C. On the distribution of \(\alpha p^ k\) modulo 1. (English) Zbl 0880.11052 Glasg. Math. J. 39, No. 2, 121-130 (1997). 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 \(\sigma= (2/3)2^{-k}\). Reviewer: M.-C.Liu (Hongkong) Cited in 1 ReviewCited in 5 Documents MSC: 11J71 Distribution modulo one 11J54 Small fractional parts of polynomials and generalizations Keywords:small fractional parts of polynomials; distribution modulo one Citations:Zbl 0504.10018; Zbl 0751.11037 PDFBibTeX XMLCite \textit{K. C. Wong}, Glasg. Math. J. 39, No. 2, 121--130 (1997; Zbl 0880.11052) Full Text: DOI References: [1] Davenport, Multiplicative number theory (1980) · Zbl 0453.10002 · doi:10.1007/978-1-4757-5927-3 [2] DOI: 10.1006/jnth.1995.1020 · Zbl 0822.11062 · doi:10.1006/jnth.1995.1020 [3] Baker, Mathematika 38 pp 170– (1991) [4] Vinogradov, C.R. Doklady, Acad.-Sci. URSS(NS) 51 pp 491– (1946) [5] Vinogradov, The method of trigonometric sums in the theory of numbers (1954) [6] Vaughan, Mathematika 24 pp 135– (1977) [7] DOI: 10.1112/plms/s3-42.2.252 · doi:10.1112/plms/s3-42.2.252 [8] DOI: 10.1112/plms/s3-72.2.241 · Zbl 0874.11052 · doi:10.1112/plms/s3-72.2.241 [9] Iwaniec, Acta Arithmetica 36 pp 171– (1980) [10] Harman, Glasgow Math. J. 24 pp 23– (1983) [11] Harman, Mathematika 28 pp 249– (1981) [12] DOI: 10.1112/jlms/s2-27.1.9 · Zbl 0504.10018 · doi:10.1112/jlms/s2-27.1.9 [13] Halberstam, Sieve methods (1974) [14] DOI: 10.1006/jnth.1993.1075 · Zbl 0786.11042 · doi:10.1006/jnth.1993.1075 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.