Dynamical Systems Seminar

We give a geometric, more constructive proof of Peck's Theorem on the simultaneous Diophantine approximation of algebraic numbers. Unlike the original argument of L.G. Peck (1961), which relies on analytic distributions of units in log-space and trace-form computations, our approach focuses on the interaction between the Minkowski embedding of a field \(K\) and the discrete dynamics of the unit group \(\mathcal{O}_{K}^{\times}\). By introducing a specific linear transformation \(A\) into the dual lattice space, we show that the approximation problem can be interpreted as a sequence of contractions and rotations within the complex embeddings of \(K\). We then show that the multiplicative action of the fundamental units produces a spiral-like trajectory that, together with Dirichlet's Approximation Theorem, drives dual lattice points into a chosen neighborhood of the approximation axes. This viewpoint gives a more direct geometric sense of the \(\log(q)\) factor and clarifies how the arithmetic structure of algebraic units influences the quality of simultaneous approximations.