## 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)].

### MSC:

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

### Keywords:

transcendence criterion; infinite products

Zbl 1175.11036
Full Text:

### 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
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