an:05666568
Zbl 1205.11082
Pupyrev, Yu. A.
Rational approximations of the number \(\root 3 \of {3}\)
EN
Math. Notes 86, No. 5, 693-703 (2009); translation from Mat. Zametki 86, No. 5, 736-747 (2009).
00257785
2009
j
11J04
Pad?? approximant; effective rational approximation; Laplace method
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 Pad?? 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 \(L_n=3^{1/3}-p_n/q_n\).
Art??ras Dubickas (Vilnius)