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

Corvaja, Pietro; Hančl, Jaroslav

Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 18, No. 3, 295-303 (2007).

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)].

Reviewer: Olaf Ninnemann (Berlin)

MSC:

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

Keywords:

transcendence criterion; infinite products

Citations:

Zbl 1175.11036

References:

[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