On irrationality measure functions for several real numbers

摘要

For an $n$-tuple $boldsymbol{alpha} = (alpha_1, ldots, alpha_n)$ of pairwise independent numbers we consider permutations [ displaylines{ boldsymbol{sigma} (t): {1, 2, 3, dots, n} rightarrow { sigma_1, sigma_2, sigma_3, ldots, sigma_n }, cr psi_{alpha_{sigma_1}}(t) psi_{alpha_{sigma_2}}(t) psi_{alpha_{sigma_3}}(t) cdots psi_{alpha_{sigma_n}}(t), } ] of irrationality measure functions $psi_alpha(t) = minnolimits_{1 leq q leq t} lVert qalpharVert$. Let $mathfrak{k}(boldsymbol{alpha})$ be the number of infinitely occurring different permutations ${boldsymbol{sigma}_1, dots, boldsymbol{sigma}_{mathfrak{k}(boldsymbol{alpha})}}$. We prove that the length of an $n$-tuple $boldsymbol{alpha}$ with $mathfrak{k}(boldsymbol{alpha}) = k$ is [ n leq frac{k(k+1)}{2} vspace{-5pt} ] and this result is optimal.