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 wellapproximable vectors on the Veronese curve and refine the best known upper bound for the exponent $\widehat{\lambda}_{n}(\zeta)$ for even $n\geq 4$.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 DOI:
 10.48550/arXiv.1704.00055
 arXiv:
 arXiv:1704.00055
 Bibcode:
 2017arXiv170400055S
 Keywords:

 Mathematics  Number Theory;
 11J13;
 11J82;
 11J83
 EPrint:
 33 pages, 2 figures