Counting the values taken by sums of \(S\)-units
Let \(K\) be a number field and let \({\mathfrak c}=(c_ 0,\ldots,c_ n)\) be \(n+1\) non-zero numbers in \(K\). Suppose \(S\) is a finite set of places of \(K\), including the archimedean ones. Denote by \(|\cdot |_ v\) the standard absolute value corresponding to a place \(v\). In this paper the counting function \(N_{\mathfrak c}(q)=\#\{x\in\mathbb P^ n(K)\); \(x\) is represented by a vector \((u_ 0,\ldots,u_ n)\) with the property \(v(u_ i)=0\) whenever \(v\not\in S\), and no subsum \(\sum c_{i_ j}u_{i_ j}\) vanishes, and \(\prod_{v\in S}| \sum c_ i u_ i|_ v<q\}\) is studied. The main result is a three term asymptotic formula for \(N_{\mathfrak c}(q)\), proven under the technical restriction that the group of \(S\)-units \(U_ S=\{u\in K; v(u)=0\) for all \(v\not\in S\}\) has a torsion-free part with a basis \(u_ 1,\ldots,u_ s\) such that for every \(v\in S\) \[ \dim_{\mathbb Q}\sum_{i}\mathbb Q\cdot \log | u_ i|_ v\geq 2. \] A crucial tool is a result of Evertse, who compared the function \(N_{\mathfrak c}(\mathfrak u)=\prod_{v\in S}| \sum c_ iu_ i|_ v\) to the height function \(H(\mathfrak u)=\prod \max_ i\{| u_ i|_ v\}\). Evertse's result is that there exist positive constants \(c_ 1\), \(c_ 2\) such that \[ c_ 1 H(\mathfrak u)^{1-\varepsilon}\leq N_{\mathfrak c}(\mathfrak u)\leq c_ 2 H(\mathfrak u). \] A technical difficulty in the current paper arises from the exponent \(1-\varepsilon\) in Evertse's estimate. From the comparison of \(N_{\mathfrak c}\) with the height function it comes as no surprise that the coefficient of the leading term \(\log (q)^{n\cdot \text{rank}\, U_ S}\) involves the regulator of \(U_ S\).
scientific article; zbMATH DE number 4148203
scientific article