Computing the Density of the Positivity Set for Linear Recurrence Sequences
The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how much more frequent are the positive entries compared to the n...
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| Vydané v: | Logical methods in computer science Ročník 19, Issue 4 |
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| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Logical Methods in Computer Science e.V
28.11.2023
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| Predmet: | |
| ISSN: | 1860-5974, 1860-5974 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | The set of indices that correspond to the positive entries of a sequence of
numbers is called its positivity set. In this paper, we study the density of
the positivity set of a given linear recurrence sequence, that is the question
of how much more frequent are the positive entries compared to the non-positive
ones. We show that one can compute this density to arbitrary precision, as well
as decide whether it is equal to zero (or one). If the sequence is
diagonalisable, we prove that its positivity set is finite if and only if its
density is zero. Further, arithmetic properties of densities are treated, in
particular we prove that it is decidable whether the density is a rational
number, given that the recurrence sequence has at most one pair of dominant
complex roots. Finally, we generalise all these results to symbolic orbits of
linear dynamical systems, thereby showing that one can decide various
properties of such systems, up to a set of density zero. |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-19(4:16)2023 |