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On some arithmetical properties of weighted sums of S-units. (English) Zbl 0738.11034

Let \(K\) be an algebraic number field and \(|\;|_ v\) (\(v\in M_ K\)) normalized absolute values on \(K\) satisfying the product formula \(\prod_ v| x|_ v=1\) for \(x\in K^*\). Further, \(S\) is a finite subset of \(M_ K\) containing all places \(v\) for which \(|\;|_ v\) is archimedean. As usual, \({\mathcal O}_ S=\{x\in K:| x|_ v\leq 1\hbox{ for } v\not\in S\}\) and \({\mathcal O}^*_ S=\{x\in K: | x|_ v=1\) for \(v\not\in S\}\) denote the ring of \(S\)- integers and multiplicative group of \(S\)-units, respectively. For \(x\in{\mathcal O}_ S\), put \(N_ S(x)=\prod_{v\in S}| x|_ v\). Let \({\mathfrak a}=(a_ 0,\ldots,a_ n)\) be a vector of non-zero elements of \({\mathcal O}_ S\) and put \(B_{n,{\mathfrak a}}=\{\beta\in{\mathcal O}_ S:\hbox{ there are }x_ 0,\ldots,x_ n\in{\mathcal O}^*_ S\hbox{ with }\beta=a_ 0x_ 0+\ldots+a_ nx_ n\}\). The authors prove some results about the set of numbers \(N_ S(\beta)\) with \(\beta\in B_{n,{\mathfrak a}}\). Denote by \(P(a)\) the greatest prime factor and by \(Q(a)\) the greatest square-free divisor of \(a\in\mathbb{Z}\), where \(P(a)=Q(a):=1\) for \(a=-1,0,1.\)
Theorem 1: For \(P>1\), the number of values \(N_ S(\beta)\) with \(\beta\in B_{n,{\mathfrak a}}\) and \(P(N_ S(\beta))\leq P\) is at most some number \(C_ 1(K,S,P,n)\) depending only on \(K,S,P,n\).
Theorem 2: Let \(n=1\). There are effectively computable numbers \(C_ 2,C_ 3\), depending only on \({\mathfrak a},K,S\) such that for every \(\beta\in B_{1,{\mathfrak a}}\) with \(N_ S(\beta)\geq C_ 2\) one has \[ Q(N_ S(\beta))\geq \exp\{C_ 3(\log\log N_ S(\beta))^ 2/\log\log\log N_ S(\beta)\}. \] The authors apply these results to obtain corollaries on recursive sequences and on decomposable form equations.
In their proof of Theorem 1, the authors use a result of K. Györy and the reviewer on the number of solutions of weighted \(S\)-unit equations [Compos. Math. 66, 329-354 (1988; Zbl 0644.10015)]. By using instead Schlickewei’s explicit bound for the number of solutions of weighted \(S\)-unit equations one can compute \(C_ 1\) explicitly. In the proof of Theorem 2, lower bounds for linear forms in ordinary as well as \(p\)-adic logarithms are used.

MSC:

11D57 Multiplicative and norm form equations
11D85 Representation problems
11D61 Exponential Diophantine equations
11D72 Diophantine equations in many variables
11J86 Linear forms in logarithms; Baker’s method

Citations:

Zbl 0644.10015
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