Rational approximations to two irrational numbers

摘要

For real $xi$ we consider the irrationality measure function $psi_xi(t) = min_{1leqslant q leqslant t, qinmathbb{Z}} | qxi${}$|$. We prove that in the case $alphapmbetanotinmathbb{Z}$ there exist arbitrary large values of $t$ with [ bigl| frac{1}{psi_alpha(t)} - frac{1}{psi_beta(t)} bigr| geqslant sqrt5bigl(mkern-2mu 1-sqrt{smash{frac{sqrt5-1}{2}}vrule width0pt height15pt depth6pt}bigr)t. ] The constant on the right-hand side is optimal.