## 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

### Citations:

Zbl 0504.10018; Zbl 0751.11037
Full Text:

### References:

 [1] Davenport, Multiplicative number theory (1980) · Zbl 0453.10002 [2] DOI: 10.1006/jnth.1995.1020 · Zbl 0822.11062 [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 [8] DOI: 10.1112/plms/s3-72.2.241 · Zbl 0874.11052 [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 [13] Halberstam, Sieve methods (1974) [14] DOI: 10.1006/jnth.1993.1075 · Zbl 0786.11042
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