## 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 $$1\not= |\alpha_1|>\max (|\alpha_2|,\ldots,|\alpha_h|)$$.

### MSC:

 11J81 Transcendence (general theory) 11J25 Diophantine inequalities 11D61 Exponential Diophantine equations

Zbl 0615.10042
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