## 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

### Citations:

Zbl 0447.10035; Zbl 0397.10026; Zbl 0504.10018
Full Text:

### References:

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