The Orbit Problem asks whether the orbit of a point under a matrix reaches a given target set. When the target is a single point, the problem was shown to be decidable in polynomial time by Kannan and Lipton. This decidability result was later extended by Chonev et al. to targets of dimension 3 (in arbitrary ambient dimension), but decidability remains open for subspaces of dimension 4. At the other extreme, the special case of the Orbit Problem in which the target set is a hyperplane of codimension 1 is equivalent to the Skolem Problem for linear recurrence sequences, whose decidability has been open for many decades. In this paper, we show that the Orbit Problem is decidable if the target subspace has dimension logarithmic in the dimension of the orbit. Over rationals, we moreover obtain a complexity bound NP^RP in this case. On the other hand, we show that the version of the Orbit Problem where the dimension of the target subspace is linear in the dimension of the orbit is as hard as the Skolem Problem.

翻译：轨道问题询问：在一个矩阵作用下，一点的轨道是否能到达给定的目标集。当目标为单点时，Kannan和Lipton证明了该问题可在多项式时间内判定。Chonev等人后来将此可判定性结果推广到维度为3的目标（在任意环境维度下），但维度为4的子空间的可判定性仍是开放问题。另一极端情况是，当目标集为余维度为1的超平面时，轨道问题的特例等价于线性递归序列的Skolem问题，其可判定性已悬而未决数十年。本文证明：若目标子空间维度相对于轨道维度呈对数增长，则轨道问题是可判定的。在有理数域上，我们还得到此情形下的复杂度界NP^RP。另一方面，我们证明：当目标子空间维度与轨道维度呈线性增长时，轨道问题的版本与Skolem问题同样困难。