A transcendence criterion for infinite products. (English) Zbl 1207.11075

The authors prove a transcendence criterion for certain infinite products of algebraic numbers. For an increasing sequence of positive integers \(a_n\) and an algebraic number \(\alpha>1\), they study the convergent infinite product \(\prod_{n}([\alpha^{a_n}]/\alpha^{a_n})\), where \([\cdot]\) denotes the integral part. It is proved in Theorem 1 that its value is transcendental, under certain hypotheses; whereas Theorem 3 shows that such hypotheses are in a sense unavoidable. This is done by applying a result of the first author and U. Zannier [Acta Math. 193, No. 2, 175–191 (2004; Zbl 1175.11036)].


11J81 Transcendence (general theory)
11J68 Approximation to algebraic numbers


Zbl 1175.11036
Full Text: DOI


[1] P. CORVAJA - U. ZANNIER, Some new applications of the Subspace Theorem . Compos. Math. 131 (2002), 319-340. 303 · Zbl 1010.11038
[2] P. CORVAJA - U. ZANNIER, On the rational approximation to the powers of an algebraic number: Solution of two problems of Mahler and Mend‘es France . Acta Math. 193 (2004), 175-191. · Zbl 1175.11036
[3] W. M. SCHMIDT, Diophantine Approximations and Diophantine Equations . Lecture Notes in Math. 1467, Springer, 1991. · Zbl 0754.11020
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