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 .
@InProceedings{luca_et_al:LIPIcs.MFCS.2022.73, author = {Luca, Florian and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{A Universal Skolem Set of Positive Lower Density}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {73:1--73:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.73}, URN = {urn:nbn:de:0030-drops-168711}, doi = {10.4230/LIPIcs.MFCS.2022.73}, annote = {Keywords: Linear Recurrence Sequences, Skolem Problem, Exponential Diophantine Equations, Sieve Methods} }
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