一句话：证明可以任意精度计算线性递推序列正性集的密度，并可判定其是否为零或一，且推广到线性动力系统的符号轨道。
本文研究线性递推序列正性集的密度，即序列中正项相对于非正项的渐近频率。作者证明该密度可以按任意精度计算，并可判定其是否等于零或一；对于可对角化序列，正性集有限当且仅当密度为零。论文还讨论密度的算术性质，特别证明在序列至多有一对主导复根时，密度是否为有理数可判定。最后，作者将上述结果推广到线性动力系统的符号轨道，表明可判定此类系统在密度零例外集下的若干性质。
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.
@article{paperbot2178,
title = {Computing the Density of the Positivity Set for Linear Recurrence Sequences},
author = {Edon Kelmendi},
journal = {Logical Methods in Computer Science},
volume = {Volume 19, Issue 4},
year = {2023},
doi = {10.46298/lmcs-19(4:16)2023}
}