Rational approximations of the number \(\root 3 \of {3}\) (Q845201)

Apparently, the best known irrationality measure of the number \(3^{1/3}\) is \(2.69267\) (due to G. Chudnovsky). This means that the inequality \(|q 3^{1/3}-p|>q^{-1.69267}\) holds for each sufficiently large positive integer \(q\) and each integer \(p\). In this paper the author shows that the constant \(1.69267\) can be replaced by the constant \(1.50308\) for a special sequence \(q=2^k\), \(k \in {\mathbb N}\), namely, the inequality \(||3^{1/3} 2^k||>0.3568^k\) holds for each sufficiently large positive integer \(k\). The proof uses Padé approximants and is effective. The improvement compared to the general estimate comes from the fact the the powers of \(2\) naturally appear in the denominators \(q_n\) of corresponding `small' linear forms \(L_n=3^{1/3}-p_n/q_n\).