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On the distribution of \(\alpha{}p^ k\) modulo one. (English) Zbl 0751.11037

The authors prove: Suppose that \(\alpha\) is irrational and \(k\geq 2\) is an integer. Then, for any real \(\beta\) and any \(\varepsilon>0\), there are infinitely many primes \(p\) satisfying \(\|\alpha p^ k+\beta\|<p^{-\lambda+\varepsilon}\) where \(\| x\|\) denotes the smallest distance from the real \(x\) to an integer and \(\lambda=3/20\) if \(k=2\), \(\lambda=(3\times 2^{k-1})^{-1}\) if \(k\geq 3\).
For \(2\leq k\leq 12\), these results improve on the previous ones due to A. Ghosh [Proc. Lond. Math. Soc., III. Ser. 42, 252-269 (1981; Zbl 0397.10026)] and the second author [J. Lond. Math. Soc., II. Ser. 27, 9-18 (1983; Zbl 0504.10018)].

MSC:

11J71 Distribution modulo one
11K06 General theory of distribution modulo \(1\)
11L07 Estimates on exponential sums
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