Note on polynomial approximation of monomials and diophantine approximation (Q1070468)

Let \(\alpha\in (0,1)\) and \[ e_{n,p}(\alpha,0):=\min_{\deg (P)\leq n}\| x^{n\alpha}-P(x)\|_{L_ p[0,1]}. \] In this note, it is shown that for \(1\leq p\leq \infty\), \[ (*)\quad \lim_{n\to \infty}e_{n,p}(\alpha,0)^{1/n}=\exp (\int^{1}_{0}\log (\frac{\alpha -x}{\alpha +x})dx), \] if and only if \(\alpha\) cannot be approximated by rationals faster than a geometric sequence, that is, for each \(\rho\in (0,1)\), there exist at most finitely many positive integers m,n such that \[ | \alpha -m/n| \leq \rho^ n. \] Further the set of \(\alpha\in (0,1)\) for which (*) fails has Hausdorff logarithmic dimension 2. The result is of interest in convergence of exponential interpolation and convergence of Padé approximants for the partial theta function.