Finding simultaneous Diophantine approximations with prescribed quality

TL;DR

This paper introduces a polynomial-time algorithm leveraging lattice basis reduction to find simultaneous Diophantine approximations with bounded Dirichlet coefficients, advancing computational methods in number theory.

Contribution

The paper presents a novel polynomial-time algorithm using the LLL-algorithm for finding simultaneous Diophantine approximations with prescribed quality.

Findings

Algorithm guarantees bounded Dirichlet coefficients depending only on dimension

Runs in polynomial time with respect to input size

Effective for high-dimensional Diophantine approximation problems

Abstract

We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses the LLL-algorithm for lattice basis reduction. We present a version of the algorithm that runs in polynomial time of the input.