An equivalence principle between polynomial and simultaneous Diophantine approximation

Abstract

We show that Mahler's classification of real numbers $\zeta$ with respect to the growth of the sequence $(w_{n}(\zeta))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(\lambda_{n}(\zeta))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximation and metric theory. We deduce estimates on the Hausdorff dimension of well-approximable vectors on the Veronese curve and refine the best known upper bound for the exponent $\widehat{\lambda}_{n}(\zeta)$ for even $n\geq 4$.