On nearly linear recurrence sequences

Shigeki Akiyama, Jan Hendrik Evertse, A. Pethő

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

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.

Original languageEnglish
Title of host publicationNumber Theory - Diophantine Problems, Uniform Distribution and Applications
Subtitle of host publicationFestschrift in Honour of Robert F. Tichy's 60th Birthday
PublisherSpringer International Publishing
Pages1-24
Number of pages24
ISBN (Electronic)9783319553573
ISBN (Print)9783319553566
DOIs
Publication statusPublished - Jun 1 2017

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Linear Recurrence
Term
Difference Sequence
Transcendental
Complex number
Finiteness
Logarithmic
Infinity
Tend
Valid
Fluctuations

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Akiyama, S., Evertse, J. H., & Pethő, A. (2017). On nearly linear recurrence sequences. In 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.

Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday. Springer International Publishing, 2017. p. 1-24.

Research output: Chapter in Book/Report/Conference proceedingChapter

Akiyama, S, Evertse, JH & Pethő, A 2017, On nearly linear recurrence sequences. in 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
Akiyama S, Evertse JH, Pethő A. On nearly linear recurrence sequences. In Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday. Springer International Publishing. 2017. p. 1-24 https://doi.org/10.1007/978-3-319-55357-3_1
Akiyama, Shigeki ; Evertse, Jan Hendrik ; Pethő, A. / On nearly linear recurrence sequences. Number Theory - Diophantine Problems, Uniform Distribution and Applications: Festschrift in Honour of Robert F. Tichy's 60th Birthday. Springer International Publishing, 2017. pp. 1-24
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