### Abstract

In a recent paper, the first and third author proved a central limit theorem for the number of coprime solutions of the Diophantine approximation problem for formal Laurent series in the setting of the classical theorem of Khintchine. In this note, we consider a more general setting and show that even an invariance principle holds, thereby improving upon earlier work of the second author. Our result yields two consequences: (i) the functional central limit theorem and (ii) the functional law of the iterated logarithm. The latter is a refinement of Khintchine's theorem for formal Laurent series. Despite a lot of research efforts, the corresponding results for Diophantine approximation of real numbers have not been established yet.

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

Pages (from-to) | 535-545 |

Number of pages | 11 |

Journal | Finite Fields and Their Applications |

Volume | 13 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2007 Jul |

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### Keywords

- Diophantine approximation
- Formal Laurent series
- Invariance principles

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Theoretical Computer Science

### Cite this

*Finite Fields and Their Applications*,

*13*(3), 535-545. https://doi.org/10.1016/j.ffa.2006.03.004

**Invariance principles for Diophantine approximation of formal Laurent series over a finite base field.** / Deligero, Eveyth; Fuchs, Michael; Nakada, Hitoshi.

Research output: Contribution to journal › Article

*Finite Fields and Their Applications*, vol. 13, no. 3, pp. 535-545. https://doi.org/10.1016/j.ffa.2006.03.004

}

TY - JOUR

T1 - Invariance principles for Diophantine approximation of formal Laurent series over a finite base field

AU - Deligero, Eveyth

AU - Fuchs, Michael

AU - Nakada, Hitoshi

PY - 2007/7

Y1 - 2007/7

N2 - In a recent paper, the first and third author proved a central limit theorem for the number of coprime solutions of the Diophantine approximation problem for formal Laurent series in the setting of the classical theorem of Khintchine. In this note, we consider a more general setting and show that even an invariance principle holds, thereby improving upon earlier work of the second author. Our result yields two consequences: (i) the functional central limit theorem and (ii) the functional law of the iterated logarithm. The latter is a refinement of Khintchine's theorem for formal Laurent series. Despite a lot of research efforts, the corresponding results for Diophantine approximation of real numbers have not been established yet.

AB - In a recent paper, the first and third author proved a central limit theorem for the number of coprime solutions of the Diophantine approximation problem for formal Laurent series in the setting of the classical theorem of Khintchine. In this note, we consider a more general setting and show that even an invariance principle holds, thereby improving upon earlier work of the second author. Our result yields two consequences: (i) the functional central limit theorem and (ii) the functional law of the iterated logarithm. The latter is a refinement of Khintchine's theorem for formal Laurent series. Despite a lot of research efforts, the corresponding results for Diophantine approximation of real numbers have not been established yet.

KW - Diophantine approximation

KW - Formal Laurent series

KW - Invariance principles

UR - http://www.scopus.com/inward/record.url?scp=33947638505&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947638505&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2006.03.004

DO - 10.1016/j.ffa.2006.03.004

M3 - Article

AN - SCOPUS:33947638505

VL - 13

SP - 535

EP - 545

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

IS - 3

ER -