## Abstract

We investigate the distribution of αp modulo one in imaginary quadratic number fields K ⊂ C with class number one, where p is restricted to prime elements in the ring of integers O = Z[ω] of K. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality ‖αp‖
_{ω} < N(p)
^{−1/8+ɛ} is satisfied for infinitely many p, where ‖ϱ‖
_{ω} measures the distance of ϱ ∈ C to O and N(p) denotes the norm of p. The proof is based on Harman’s sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

Original language | English |
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Pages (from-to) | 719-760 |

Number of pages | 42 |

Journal | Journal de Théorie des Nombres de Bordeaux |

Volume | 32 |

Issue number | 3 |

DOIs | |

Publication status | Published - 8 Jan 2021 |

## Keywords

- Diophantine approximation
- Distribution modulo one
- Imaginary quadratic field
- Poisson summation
- Smoothed sum

## ASJC Scopus subject areas

- Algebra and Number Theory