×

Subspace Lang conjecture and some remarks on a transcendental criterion. (English) Zbl 1432.11078

Summary: Let \(b\geq 2\) be an integer and \(\alpha \) is a non-zero real number written in \(b\)-ary expansion. In [C. R., Math., Acad. Sci. Paris 339, No. 1, 11–14 (2004; Zbl 1119.11019)], B. Adamczewski et al. provided a criterion for an irrational number to be a transcendental number using \(b\)-ary expansion. In this paper, we make some remarks on this criterion and, under the assumption of subspace Lang conjecture, we extend this criterion for a much wider class of irrational numbers.

MSC:

11J68 Approximation to algebraic numbers
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11J87 Schmidt Subspace Theorem and applications

Citations:

Zbl 1119.11019
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adamczewski, B; Bugeaud, Y, On the complexity of algebraic numbers - I, expansions in integer bases, Ann. Math., 165, 547-565, (2007) · Zbl 1195.11094 · doi:10.4007/annals.2007.165.547
[2] Adamczewski, B; Bugeaud, Y; Luca, F, Sur la complexité des nombres algébriques, C. R. Acad. Sci. Paris, 339, 11-14, (2004) · Zbl 1119.11019 · doi:10.1016/j.crma.2004.04.012
[3] Dixit A B, Rath P and Shankar A, Lang’s conjecture and complexity of algebraic numbers, preprint, http://www.cmi.ac.in/ adixit/Langborel.pdf · Zbl 0079.27401
[4] Ridout, D, Rational approximations to algebraic numbers, Mathematika, 4, 125-131, (1957) · Zbl 0079.27401 · doi:10.1112/S0025579300001182
[5] Roth, KF, Rational approximations to algebraic numbers, Mathematika, 2, 1-20, (1955) · Zbl 0064.28501 · doi:10.1112/S0025579300000644
[6] Schlickewei, HP, The \(\mathfrak{p}\)-adic thue-Siegel-Roth-Schmidt theorem, Arch. Math. (Basel), 29, 267-270, (1977) · Zbl 0365.10026 · doi:10.1007/BF01220404
[7] Schmidt W M, Diophantine Approximation, Lecture Notes in Mathematics 785 (1980) (Springer)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.