Sums of \(S\)-units in recurrence sequences
Let \( S=\{p_1, \ldots, p_s\} \) be a set of primes, and write \( \mathbb{Z}_S \) for the set of those integers which have no prime divisors outside \( S \). Let \( U_n \) be a linear recurrence sequence of integers. A linear recurrence sequence of order \(r\) is the sequence \( (U_n)_{n\ge 0} \) satisfying the relation \begin{align*} U_n=a_1U_{n-1}+\cdots+a_rU_{n-r}, \end{align*} where \( a_1, \ldots, a_r \) are integers with \( a_r\ne 0 \) and \( U_0, \ldots, U_{r-1} \) are integrs not all zero. The characteristic polynomial of \( U_n \) is defined by \begin{align*} f(x):=x^r-a_1x^{r-1}-\cdots-a_r=\prod_{i=1}^{t}(x-\alpha_i)^{m_i}, \end{align*} where \( \alpha_1, \ldots, \alpha_t \) are distinct algebraic numbers and \( m_1, \ldots, m_t \) are integers. The sequence \( (U_n)_{n\ge 0} \) is called \textit{degenerate} if there are integers \( i, j \) with \( 1\le i<j\le t \) such that \( \alpha_i/\alpha_j \) is not a root of unity; otherwise it is called \textit{non-degenerate}. If for some \( i \) with \( 1\le i\le t \) we have \( |\alpha_i|>|\alpha_j| \) for all \( j \) with \( 1\le j\le t \) and \( j\ne i \), then \( \alpha_i \) is called a \textit{dominant root} of the sequence \( (U_n)_{n\ge 0} \). In the paper under review, the authors study the Diophantine equation \[U_n=w_1+\cdots+w_k \tag{1}\] with some arbitrary but fixed \( k\ge 1 \), in unknown \( n\ge 0 \) and \( w_1, \ldots, w_k \in \mathbb{Z}_S \). Their major aim is to establish finiteness results for the solutions of (1). Their main results are the following. Theorem 2. Let \(U_n\) be a non-degenerate recurrence sequence of order \(r\), and write \(f (x)\) for the characteristic polynomial of \(U_n\). Suppose that at least one of the following two conditions is valid: \begin{itemize} \item[(i)]\( f(x) \) has at least one irrational root, \item[(ii)]\( f(0) \) has a prime divisor outside \( S \). \end{itemize} Then for any fixed \(k \ge 1\), equation (1) is solvable at most for finitely many \(n\). Further, the number of indices \(n\) for which (1) is solvable for this fixed \(k\), can be bounded by an effectively computable constant depending only on \(r\), \(s\), and \(k\). Theorem 3. Let \((U_n )_{n\ge 0}\) be a non-degenerate recurrence sequence of order \(r \ge 2\). Suppose that \((U_n )_{n\ge 0}\) has a dominant root which is not an integer. Let \(\varepsilon > 0\) be arbitrary. Then for all \(k \ge 1\), for all solutions \(n, w_1, \ldots, w_k\) of (1) satisfying \(|w_i |^{1+\varepsilon} < |w_k |\) \((i = 1, \ldots , k - 1)\) we have \begin{align*} \max(n, |w_1|, \ldots, |w_k|)\le C, \end{align*} where \( C \) is an effectively computable constant depending only on \( \varepsilon, U_n, r, s, p_1, \ldots, p_s, k \). The proofs of Theorems 2 and 3 rely on a clever combination of techniques and results in number theory, the usual properties of non-degenerate linear recurrence sequences, and the theory of nonzero linear forms in logarithms of algebraic numbers á la Baker.
scientific article; zbMATH DE number 6987939
scientific article