### Abstract

A nearly linear recurrence sequence (nlrs) is a complex sequence (a_{n}) with the property that there exist complex numbers A_{0},...., A_{d-1} such that the sequence is bounded. We give an asymptotic Binet-type formula for such sequences. We compare (a_{n}) with a natural linear recurrence sequence (lrs) (ã_{n}) associated with it and prove under certain assumptions that the difference sequence (a_{n} - ã_{n}) tends to infinity. We show that several finiteness results for lrs, in particular the Skolem-Mahler-Lech theorem and results on common terms of two lrs, are not valid anymore for nlrs with integer terms. Our main tool in these investigations is an observation that lrs with transcendental terms may have large fluctuations, quite different from lrs with algebraic terms. On the other hand, we show under certain hypotheses that though there may be infinitely many of them, the common terms of two nlrs are very sparse. The proof of this result combines our Binet-type formula with a Baker type estimate for logarithmic forms.

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

Title of host publication | Number Theory - Diophantine Problems, Uniform Distribution and Applications |

Subtitle of host publication | Festschrift in Honour of Robert F. Tichy's 60th Birthday |

Publisher | Springer International Publishing |

Pages | 1-24 |

Number of pages | 24 |

ISBN (Electronic) | 9783319553573 |

ISBN (Print) | 9783319553566 |

DOIs | |

Publication status | Published - Jun 1 2017 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday*(pp. 1-24). Springer International Publishing. https://doi.org/10.1007/978-3-319-55357-3_1

**On nearly linear recurrence sequences.** / Akiyama, Shigeki; Evertse, Jan Hendrik; Pethő, A.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday.*Springer International Publishing, pp. 1-24. https://doi.org/10.1007/978-3-319-55357-3_1

}

TY - CHAP

T1 - On nearly linear recurrence sequences

AU - Akiyama, Shigeki

AU - Evertse, Jan Hendrik

AU - Pethő, A.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - A nearly linear recurrence sequence (nlrs) is a complex sequence (an) with the property that there exist complex numbers A0,...., Ad-1 such that the sequence is bounded. We give an asymptotic Binet-type formula for such sequences. We compare (an) with a natural linear recurrence sequence (lrs) (ãn) associated with it and prove under certain assumptions that the difference sequence (an - ãn) tends to infinity. We show that several finiteness results for lrs, in particular the Skolem-Mahler-Lech theorem and results on common terms of two lrs, are not valid anymore for nlrs with integer terms. Our main tool in these investigations is an observation that lrs with transcendental terms may have large fluctuations, quite different from lrs with algebraic terms. On the other hand, we show under certain hypotheses that though there may be infinitely many of them, the common terms of two nlrs are very sparse. The proof of this result combines our Binet-type formula with a Baker type estimate for logarithmic forms.

AB - A nearly linear recurrence sequence (nlrs) is a complex sequence (an) with the property that there exist complex numbers A0,...., Ad-1 such that the sequence is bounded. We give an asymptotic Binet-type formula for such sequences. We compare (an) with a natural linear recurrence sequence (lrs) (ãn) associated with it and prove under certain assumptions that the difference sequence (an - ãn) tends to infinity. We show that several finiteness results for lrs, in particular the Skolem-Mahler-Lech theorem and results on common terms of two lrs, are not valid anymore for nlrs with integer terms. Our main tool in these investigations is an observation that lrs with transcendental terms may have large fluctuations, quite different from lrs with algebraic terms. On the other hand, we show under certain hypotheses that though there may be infinitely many of them, the common terms of two nlrs are very sparse. The proof of this result combines our Binet-type formula with a Baker type estimate for logarithmic forms.

UR - http://www.scopus.com/inward/record.url?scp=85033706384&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85033706384&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-55357-3_1

DO - 10.1007/978-3-319-55357-3_1

M3 - Chapter

AN - SCOPUS:85033706384

SN - 9783319553566

SP - 1

EP - 24

BT - Number Theory - Diophantine Problems, Uniform Distribution and Applications

PB - Springer International Publishing

ER -