A Universal Skolem Set of Positive Lower Density

Authors Florian Luca , Joël Ouaknine , James Worrell

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Florian Luca
  • School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
  • Research Group in Algebraic Structures & Applications, King Abdulaziz University, Saudi Arabia
  • Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
James Worrell
  • Department of Computer Science, University of Oxford, UK

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Florian Luca, Joël Ouaknine, and James Worrell. A Universal Skolem Set of Positive Lower Density. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 73:1-73:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The Skolem Problem asks to decide whether a given integer linear recurrence sequence (LRS) has a zero term. Decidability of this problem has been open for many decades, with little progress since the 1980s. Recently, a new approach was initiated via the notion of a Skolem set - a set of positive integers relative to which the Skolem Problem is decidable. More precisely, 𝒮 is a Skolem set for a class ℒ of integer LRS if there is an effective procedure that, given an LRS in ℒ, decides whether the sequence has a zero in 𝒮. A recent work exhibited a Skolem set for the class of all LRS that, while infinite, had density zero. In the present work we construct a Skolem set of positive lower density for the class of simple LRS .

Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic algorithms
  • Computing methodologies → Number theory algorithms
  • Linear Recurrence Sequences
  • Skolem Problem
  • Exponential Diophantine Equations
  • Sieve Methods


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  1. S. Akshay, N. Balaji, A. Murhekar, R. Varma, and N. Vyas. Near-optimal complexity bounds for fragments of the Skolem problem. In 37th International Symposium on Theoretical Aspects of Computer Science, STACS, volume 154 of LIPIcs, pages 37:1-37:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  2. F. Amoroso and E. Viada. Small points on subvarieties of a torus. Duke Mathematical Journal, 150(3), 2009. Google Scholar
  3. A. Baker. A Comprehensive Course in Number Theory. Cambridge University Press, 2012. Google Scholar
  4. D. Beauquier, A. M. Rabinovich, and A. Slissenko. A logic of probability with decidable model checking. J. Log. Comput., 16(4), 2006. Google Scholar
  5. J. Berstel and C. Reutenauer. Noncommutative Rational Series with Applications. Cambridge University Press, 2010. Google Scholar
  6. V. Blondel and J. Tsitsiklis. A survey of computational complexity results in systems and control. Automatica, 36(9):1249-1274, 2000. URL: https://doi.org/10.1016/S0005-1098(00)00050-9.
  7. J.-Y. Cai, R. J. Lipton, and Y. Zalcstein. The complexity of the A B C problem. SIAM J. Comput., 29(6), 2000. Google Scholar
  8. G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward. Recurrence Sequences. American Mathematical Society, 2003. Google Scholar
  9. J.-H. Evertse. On sums of s-units and linear recurrences. Compositio Mathematica, 53(2):225-244, 1984. Google Scholar
  10. N. Fijalkow, J. Ouaknine, A. Pouly, J. Sousa Pinto, and J. Worrell. On the decidability of reachability in linear time-invariant systems. In Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control, HSCC, pages 77-86. ACM, 2019. Google Scholar
  11. H. Halberstam and H.-E. Richert. Sieve methods. LMS Monographs. Academic Press, 1974. Google Scholar
  12. G. H. Hardy and E. M. Wright. An introduction to the theory of numbers. Oxford, at the Clarendon Press, 1954. 3rd ed. Google Scholar
  13. R. Kannan and R. J. Lipton. Polynomial-time algorithm for the orbit problem. JACM, 33(4), 1986. Google Scholar
  14. C. Lech. A note on recurring series. Ark. Mat., 2, 1953. Google Scholar
  15. F. Luca, J. Ouaknine, and J. Worrell. Universal Skolem Sets. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS, pages 1-6. IEEE, 2021. Google Scholar
  16. K. Mahler. Eine arithmetische Eigenschaft der Taylor Koeffizienten rationaler Funktionen. Proc. Akad. Wet. Amsterdam, 38, 1935. Google Scholar
  17. D. W. Masser. Linear relations on algebraic groups. In New Advances in Transcendence Theory. Cambridge University Press, 1988. Google Scholar
  18. M. Mignotte, T. Shorey, and R. Tijdeman. The distance between terms of an algebraic recurrence sequence. J. für die reine und angewandte Math., 349, 1984. Google Scholar
  19. J. Ouaknine and J. Worrell. On linear recurrence sequences and loop termination. ACM SIGLOG News, 2(2):4-13, 2015. Google Scholar
  20. J. Piribauer and C. Baier. On Skolem-hardness and saturation points in markov decision processes. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, volume 168 of LIPIcs, pages 138:1-138:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  21. G. Rozenberg and A. Salomaa. Cornerstones of Undecidability. Prentice Hall, 1994. Google Scholar
  22. H.P. Schlickewei and W.P. Schmidt. The number of solutions of polynomial-exponential equations. Compositio Mathematica, 120:193-225, January 2000. Google Scholar
  23. T. Skolem. Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen. In Comptes rendus du congrès des mathématiciens scandinaves, 1934. Google Scholar
  24. T. Tao. Structure and randomness: pages from year one of a mathematical blog. American Mathematical Society, 2008. Google Scholar
  25. A. J. Van Der Poorten and H. P. Schlickewei. Additive relations in fields. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 51(1):154-170, 1991. Google Scholar
  26. N. K. Vereshchagin. The problem of appearance of a zero in a linear recurrence sequence (in Russian). Mat. Zametki, 38(2), 1985. Google Scholar