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