The arithmetic version of Anderson localization (AL), i.e., AL with explicit arithmetic description on both the localization frequency and the localization phase, was first given by Jitomirskaya (Ann Math 150:1159–1175, 1999) for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) to a class of one dimensional quasi-periodic long-range operators. In this paper, we propose a novel approach based on an arithmetic version of Aubry duality and quantitative reducibility. Our method enables us to prove the same result for the class of quasi-periodic long-range operators in all dimensions, which includes Jitomirskaya (Ann Math 150:1159–1175, 1999) and Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) as special cases.

Ji.mirskayaskaya（Ann Math 150：1159-1175，1999）首先为几乎Mathieu算子（AMO）给出了安德森本地化（AL）的算术版本，即在本地化频率和本地化阶段均具有明确算术描述的AL。 ）。后来，结果由Bourgain和Jitomirskaya进行了推广（Invent Math 148：453-463，2002），是一类一维的准周期远程算子。在本文中，我们提出了一种基于Aubry对偶和定量约简算法的新颖方法。我们的方法使我们能够证明所有维度上的准周期远程算子的结果相同，其中包括Jitomirskaya（Ann Math 150：1159-1175，1999）和Bourgain and Jitomirskaya（Invent Math 148：453-463，2002）作为特例。