##
**Some new applications of the subspace theorem.**
*(English)*
Zbl 1010.11038

Consider the gap series (1) \(f(z)=\sum_{k=0}^{\infty} a_kz^{m_k}\) where the numbers \(a_k\) are non-zero elements of an algebraic number field \(K\subset \mathbb{C}\) and where \(\{ m_k\}\) is a strictly increasing sequence of positive integers. Suppose that \(f\) has radius of convergence \(R>0\). Let \(S=[K:{\mathbb{Q}}]\). Assume that
\[
\lim_{k\to\infty} \big(m_k+S(\sum_{i=1}^k h(a_i)\big)/m_{k+1}=0,\tag{2}
\]
where \(h(\alpha)\) denotes the absolute logarithmic height of an algebraic number \(\alpha \). In 1973, Cijsouw and Tijdeman showed that if \(\alpha\) is algebraic and \(0< |\alpha |<R\) then \(f(\alpha)\) is transcendental. In 1980, Bundschuh and Wylegala showed that if \(\alpha_1,\ldots,\alpha_n\) are algebraic numbers with \(0<|\alpha_1 |<|\alpha_2 |<\cdots <|\alpha_n|<R\), then \(f(\alpha_1),\ldots, f(\alpha_n)\) are algebraically independent. In 1987, K. Nishioka [Compos. Math. 62, 53-61 (1987; Zbl 0615.10042)] proved a considerable extension of this result by applying Schmidt’s Subspace Theorem. In fact, for given algebraic numbers \(\alpha_1,\ldots,\alpha_n\) she gave a necessary and sufficient condition, completely explicit in terms of \(\alpha_1,\ldots,\alpha_n\) such that the set \(\{ f^{(l)}(\alpha_i): i=1,\ldots,n, l\geq 0\}\) is algebraically independent. In 1986, she had proved a \(p\)-adic analogue of the result of Bundschuh and Wylegala by \(p\)-adic methods.

In the present paper, the authors prove transcendence results for numbers \(f(\alpha)\) where \(\alpha\) is an algebraic number and \(f\) a power series as above, but satisfying a weaker condition instead of (2). To formulate this weaker condition is rather complicated, but it amounts to saying that there are a number \(L>1\) in terms of \(\alpha\) and an integer \(N\geq 1\) such that \(m_{k+N}/m_k>L\) for all sufficiently large \(k\). The authors prove results both in the archimedean and non-archimedean case.

Apart from these transcendence results, the authors deduce some results for Laurent series and for certain linear recurrence sequences. For instance, they show that if \(f\) is a Laurent series with complex algebraic coefficients (not necessarily generating a number field) converging for \(0<|z|<1\) and if \(q\) is a complex algebraic number with \(0<|q|<1\) such that \(f(q^n)\in{\mathbb{Z}}\) for infinitely many \(n\), then \(f\) is a Laurent polynomial.

One may formulate a general conjecture that if \(\{z_n\}\) is a linear recurrence sequence with elements from a number field \(K\) and if \(g\in K[X,Z]\) is a non-zero polynomial such that \(g(z_n,X)=0\) has a solution in \(K\) for infinitely many \(n\), then there are an arithmetic progression \({\mathcal P}\), and a linear recurrence sequence \(\{ x_n\}\) with entries in \(K\), such that \(g(z_n,x_n)=0\) for all \(n\in{\mathcal P}\). The authors prove this for sequences \(\{ z_n\}\) of the special shape \(z_n=\sum_{j=1}^h c_j\alpha_j^n\), where the \(c_j,\alpha_j\) belong to \(K\) and \(1\not= |\alpha_1|>\max (|\alpha_2|,\ldots,|\alpha_h|)\).

In the present paper, the authors prove transcendence results for numbers \(f(\alpha)\) where \(\alpha\) is an algebraic number and \(f\) a power series as above, but satisfying a weaker condition instead of (2). To formulate this weaker condition is rather complicated, but it amounts to saying that there are a number \(L>1\) in terms of \(\alpha\) and an integer \(N\geq 1\) such that \(m_{k+N}/m_k>L\) for all sufficiently large \(k\). The authors prove results both in the archimedean and non-archimedean case.

Apart from these transcendence results, the authors deduce some results for Laurent series and for certain linear recurrence sequences. For instance, they show that if \(f\) is a Laurent series with complex algebraic coefficients (not necessarily generating a number field) converging for \(0<|z|<1\) and if \(q\) is a complex algebraic number with \(0<|q|<1\) such that \(f(q^n)\in{\mathbb{Z}}\) for infinitely many \(n\), then \(f\) is a Laurent polynomial.

One may formulate a general conjecture that if \(\{z_n\}\) is a linear recurrence sequence with elements from a number field \(K\) and if \(g\in K[X,Z]\) is a non-zero polynomial such that \(g(z_n,X)=0\) has a solution in \(K\) for infinitely many \(n\), then there are an arithmetic progression \({\mathcal P}\), and a linear recurrence sequence \(\{ x_n\}\) with entries in \(K\), such that \(g(z_n,x_n)=0\) for all \(n\in{\mathcal P}\). The authors prove this for sequences \(\{ z_n\}\) of the special shape \(z_n=\sum_{j=1}^h c_j\alpha_j^n\), where the \(c_j,\alpha_j\) belong to \(K\) and \(1\not= |\alpha_1|>\max (|\alpha_2|,\ldots,|\alpha_h|)\).

Reviewer: Jan-Hendrik Evertse (Leiden)

### MSC:

11J81 | Transcendence (general theory) |

11J25 | Diophantine inequalities |

11D61 | Exponential Diophantine equations |