发表机构
Institut de Recherche en Informatique Fondamentale（基础信息学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出伪有效方法，证明Skolem-Mahler-Lech定理的有限性在低高度扰动下仍成立，并为线性轨道建立移动目标有限性结果。
AI 中文摘要
Skolem-Mahler-Lech定理断言，非退化线性递推序列（LRS）只有有限多个零点，但该定理是无效的，因为目前没有已知的一般程序来确定这些零点。本文引入了一种“伪有效”的方法：与其为每个序列寻求有效的界，不如在自然等价意义下，获得对所有但受控数量的有界高度序列都成立的显式界。对于阶数为3的LRS，本文建立了这样的结果，并在特征根满足适当高度约束的条件下，将其推广到所有更高阶的情况。其底层机制是对于受低高度乘法扰动影响的指数-多项式方程的定量理论。特别地，本文证明了Skolem-Mahler-Lech定理背后的有限性现象在高度允许以足够小的速率线性增长的算术扰动下仍然成立。作为进一步的应用，本文为线性轨道进入真线性子空间的扩张算术邻域建立了一个移动目标有限性结果。
英文摘要
The Skolem--Mahler--Lech Theorem asserts that a non-degenerate linear recurrence sequence (LRS) has only finitely many zeros, but is ineffective in the sense that no general procedure is known to determine them. In this paper a \emph{pseudo-effective} approach is introduced: rather than seeking effective bounds for every sequence, one obtains explicit bounds valid for all but a controlled number of sequences of bounded height, up to a natural equivalence. Such results are established for order $3$ LRS's, and extended to all higher orders subject to suitable height constraints on the characteristic roots. The underlying mechanism is a quantitative theory for exponential-polynomial equations subject to low-height multiplicative perturbations. In particular, it is shown that the finiteness phenomenon underlying the Skolem--Mahler--Lech Theorem persists under arithmetic perturbations whose height is permitted to grow linearly at a sufficiently small rate. As a further application, a moving-target finiteness result is established for linear orbits entering expanding arithmetic neighbourhoods of proper linear subspaces.
