线性递推序列正性集的密度计算
Computing the Density of the Positivity Set for Linear Recurrence Sequences
AI总结：
本文研究线性递推序列正性集的密度计算问题，证明可任意精度计算密度、判定密度是否为0/1及是否为有理数等，还将结果推广至线性动力系统的符号轨道。
摘要：
对应于数列正项的索引集合被称为该序列的正性集。本文研究给定线性递推序列的正性集密度，即正项相较于非正项的出现频率更高的问题。我们证明可以以任意精度计算该密度，还能判定其是否等于零（或一）。若序列可对角化，我们证明其正性集有限当且仅当密度为零。此外，本文还研究了密度的算术性质，特别证明了在递推序列至多有一对主导复根的情况下，密度是否为有理数是可判定的。最后，我们将所有这些结果推广到线性动力系统的符号轨道，从而证明可以判定这类系统的多种性质，仅在一个密度为零的集合上存在例外。
英文摘要：
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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