On Nearly Linear Recurrence Sequences

  • Shigeki Akiyama
  • Jan-Hendrik Evertse
  • Attila PethőEmail author


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 \(\big(a_{n+d} + A_{d-1}a_{n+d-1} + \cdots + A_{0}a_{n}\big)_{n=0}^{\infty }\) is bounded. We give an asymptotic Binet-type formula for such sequences. We compare (a n ) with a natural linear recurrence sequence (lrs) \((\tilde{a}_{n})\) associated with it and prove under certain assumptions that the difference sequence \((a_{n} -\tilde{ a}_{n})\) 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.

2010 Mathematics Subject Classification:




Research supported in part by the OTKA grants NK104208, NK115479.


  1. 1.
    S. Akiyama, A. Pethő, Discretized rotation has infinitely many periodic orbits. Nonlinearity 26, 871–880 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, J. Thuswaldner, Generalized radix representations and dynamical systems I. Acta Math. Hungar. 108(3), 207–238 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. Akiyama, H. Brunotte, A. Pethő, W. Steiner, Remarks on a conjecture on certain integer sequences. Period. Math. Hung. 52, 1–17 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. Akiyama, H. Brunotte, A. Pethő, W. Steiner, Periodicity of certain piecewise affine planar maps. Tsukuba J. Math. 32(1), 1–55 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    J.-H. Evertse, On sums of S-units and linear recurrences. Compos. Math. 53, 225–244 (1984)MathSciNetzbMATHGoogle Scholar
  6. 6.
    G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 4th edn. (with corrections) (Clarendon Press, Oxford, 1975)Google Scholar
  7. 7.
    I. Lánczi, P. Turán, Számelmélet (Number Theory) (Tankönyvkiadó, Budapest, 1969) [in Hungarian]Google Scholar
  8. 8.
    M. Laurent, Equations exponentielles polynômes et suites récurrentes linéaires II. J. Number Theory 31, 24–53 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    C. Lech, A note on recurring series. Ark. Mat. 2, 417–421 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J.H. Lowenstein, S. Hatjispyros, F. Vivaldi, Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off. Chaos 7, 49–56 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    E.M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II. Izv. Ross. Akad. Nauk Ser. Mat. 64(6), 125–180 (2000). Translation in Izv. Math. 64(6), 1217–1269 (2000)Google Scholar
  12. 12.
    M. Mignotte, Intersection des images de certaines suites récurrentes linéaires. Theor. Comput. Sci. 7, 117–122 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Mignotte, T.N. Shorey, R. Tijdeman, The distance between terms of an algebraic recurrence sequence. J. Reine Angew. Math. 349, 63–76 (1984)MathSciNetzbMATHGoogle Scholar
  14. 14.
    A. Pethő, Notes on CNS polynomials and integral interpolation, in More Sets, Graphs and Numbers, ed. by E. Győry, G.O.H. Katona, L. Lovász. Bolyai Society Mathematical Studies, vol. 15 (Springer, Berlin, 2006), pp. 301–315Google Scholar
  15. 15.
    A. Pethő, P. Varga, Canonical number systems over imaginary quadratic euclidean domains. Colloq. Math. 146, 165–186 (2017)CrossRefzbMATHGoogle Scholar
  16. 16.
    A.J. van der Poorten, H.P. Schlickewei, The Growth Conditions for Recurrence Sequences. Macquarie University, NSW, Australia, Report 82.0041 (1982)Google Scholar
  17. 17.
    R. Salem, Algebraic Numbers and Fourier Analysis (D. C. Heath and Co., Boston, MA, 1963)zbMATHGoogle Scholar
  18. 18.
    T.N. Shorey, R. Tijdeman, Exponential Diophantine Equations. Cambridge Tracts in Mathematics, vol. 87 (Cambridge University Press, Cambridge, 1986)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Shigeki Akiyama
    • 1
  • Jan-Hendrik Evertse
    • 2
  • Attila Pethő
    • 3
    Email author
  1. 1.Institute of MathematicsUniversity of TsukubaTsukubaJapan
  2. 2.Mathematical InstituteLeiden UniversityRA LeidenThe Netherlands
  3. 3.Department of Computer ScienceUniversity of DebrecenDebrecenHungary

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