### Abstract

We consider a Kurzweil type inhomogeneous Diophantine approximation theorem in the field of the formal Laurent series for a monotone sequence of approximation. We find a necessary and sufficient condition for irrational f and monotone increasing (^{ℓn}) that there are infinitely many polynomials P and Q such that |Qf-P-g|<q-^{n}-^{ℓn}, n=deg(Q) for almost every g. We also study some conditions for irrational f such that for all monotone increasing (^{ℓn}) with Σq- ^{ℓn}=∞ there are infinitely many solutions for almost every g.

Original language | English |
---|---|

Pages (from-to) | 64-75 |

Number of pages | 12 |

Journal | Finite Fields and Their Applications |

Volume | 20 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Mar |

### Fingerprint

### Keywords

- Formal Laurent series
- Inhomogeneous Diophantine approximation
- Kurzweil type theorem

### ASJC Scopus subject areas

- Applied Mathematics
- Algebra and Number Theory
- Theoretical Computer Science
- Engineering(all)

### Cite this

*Finite Fields and Their Applications*,

*20*(1), 64-75. https://doi.org/10.1016/j.ffa.2012.12.002

**A refined Kurzweil type theorem in positive characteristic.** / Kim, Dong Han; Nakada, Hitoshi; Natsui, Rie.

Research output: Contribution to journal › Article

*Finite Fields and Their Applications*, vol. 20, no. 1, pp. 64-75. https://doi.org/10.1016/j.ffa.2012.12.002

}

TY - JOUR

T1 - A refined Kurzweil type theorem in positive characteristic

AU - Kim, Dong Han

AU - Nakada, Hitoshi

AU - Natsui, Rie

PY - 2013/3

Y1 - 2013/3

N2 - We consider a Kurzweil type inhomogeneous Diophantine approximation theorem in the field of the formal Laurent series for a monotone sequence of approximation. We find a necessary and sufficient condition for irrational f and monotone increasing (ℓn) that there are infinitely many polynomials P and Q such that |Qf-P-g|n-ℓn, n=deg(Q) for almost every g. We also study some conditions for irrational f such that for all monotone increasing (ℓn) with Σq- ℓn=∞ there are infinitely many solutions for almost every g.

AB - We consider a Kurzweil type inhomogeneous Diophantine approximation theorem in the field of the formal Laurent series for a monotone sequence of approximation. We find a necessary and sufficient condition for irrational f and monotone increasing (ℓn) that there are infinitely many polynomials P and Q such that |Qf-P-g|n-ℓn, n=deg(Q) for almost every g. We also study some conditions for irrational f such that for all monotone increasing (ℓn) with Σq- ℓn=∞ there are infinitely many solutions for almost every g.

KW - Formal Laurent series

KW - Inhomogeneous Diophantine approximation

KW - Kurzweil type theorem

UR - http://www.scopus.com/inward/record.url?scp=84873151037&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84873151037&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2012.12.002

DO - 10.1016/j.ffa.2012.12.002

M3 - Article

AN - SCOPUS:84873151037

VL - 20

SP - 64

EP - 75

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

IS - 1

ER -