SIMULTANEOUS RATIONAL APPROXIMATIONS OF p-ADIC NUMBERS BY THE LLL LATTICE BASIS REDUCTION ALGORITHM

In this paper we construct multi-dimensional p-adic approximation lattices by simultaneous rational approximations of p-adic numbers. For analyzing these p-adic lattices we apply the LLL algorithm due to Lenstra, Lenstra and Lovász, which has been widely used to solve the various NP problems such as SVP (Shortest Vector Problems), ILP (Integer Linear Programing) .. and so on. In a twodimensional lattice the Gauss reduction algorithm for finding the shortest vector is most powerful and useful. The LLL algorithm, which is a multi-dimensional extension of the Gauss algorithm, approximately solves SVP within a factor of 2 for the lattice dimension n(≥ 3) in polynomial times. Using the open source software SAGE, we compare the minimum norms of the vectors given by the LLL reduction algorithm and the norms of vectors estimated by the simultaneous approximation theory. We also study the two types of simultaneous approximations of p-adic numbers, which can be transferred from one of types to the other type by the famous Transference Principle. The Transference Principle only gives the equivalence relation between these two types on the existence of solutions of approximation inequalities. Any algorithms, which give the constructive relations between these two types of solutions, have not yet been known. Here we can give this algorithm by using LLL reduction algorithms.