Measures associated to the inverse regulator of a number field (Q1202429)

Let \(K\) be a totally real algebraic number field of degree \(r+1\) over \(\mathbb Q\). Write \({\mathcal O}_ K\) for the ring of integers of \(K\) and \(U_ K\) for the group of units of \({\mathcal O}_ K\). For \(U_ K/\{\pm 1\}\) and taking the logarithm of the absolute values of units, \(r+1\) linear forms are obtained which are called \(\varphi_ j(x)\), \(1\leq j\leq r+1\). Let \(T_ q\) be the region defined by \(T_ q=\{x\in\mathbb R^ r:\varphi_ j(x)\leq\log q\), \(j=1,\dots,r+1\}\). Let \(J_ q\) denote the maximal collection of \(r\)-dimensional unit cubes, with lattice point centres, strictly contained in \(T_ q\). It is proved in this paper that \[ \mu_{r-1}(\partial J_ q)=2\mu_ r(T_ q-J_ q)+o((\log q)^{r-1}), \] as \(q\to\infty\), where \(\mu_ k\) denotes \(k\)-dimensional Lebesgue measure in \(\mathbb R^ r\) and \(\partial J_ q\) denotes the boundary of \(J_ q\).