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：轨道问题研究一个点在矩阵作用下的轨道是否可达给定目标集。当目标为单点时，Kannan与Lipton已证明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至三维目标集（在任意环境维度中），但对于四维子空间的可判定性仍悬而未决。在另一极端情形中，当目标集为余维1的超平面时，轨道问题的特例等价于线性递推序列的Skolem问题——其可判定性已悬置数十年。本文证明：若目标子空间的维度相对于轨道维度呈对数规模，则轨道问题可判定。在有理数域情形下，我们进一步获得NP^RP复杂度界。另一方面，我们证明当目标子空间维度相对于轨道维度呈线性规模时，该版本轨道问题具有与Skolem问题同等的难度。