Let \(X\) be a subvariety, and \(\Gamma\) a finitely generated subgroup, in \({\mathbb G}_m^N(\overline{\mathbb Q})\cong(\overline{\mathbb Q}^*)^N\). One considers the division group \(\overline\Gamma\) (given by \(x\in\overline\Gamma\) if \(x^m\in\Gamma\) for some \(m>0\)) and its associated sets \(X\cap\overline\Gamma_\varepsilon\) and \(X\cap C(\overline\Gamma,\varepsilon)\) in \({\mathbb G}_m^N(\overline{\mathbb Q})\). These are defined in terms of the Weil height \(h\) on \({\mathbb G}_m^N(\overline{\mathbb Q})\) by \(\overline\Gamma_\varepsilon=\{yz\mid y\in\overline\Gamma,\;h(z)<\varepsilon\}\) and \(C(\overline\Gamma,\varepsilon)=\{yz\mid y\in\overline\Gamma,\;h(z)<\varepsilon(1+h(y))\}\). The aim is to understand in detail the intersections of these sets with \(X\), for certain special \(X\).
By work of Poonen and others it is known that there exists an \(\varepsilon\) such that \(X\cap \overline\Gamma_\varepsilon\) is contained in a finite union \(x_1H_1\cup\cdots\cup x_tH_t\) of cosets of algebraic subgroups of \({\mathbb G}_m^N(\overline{\mathbb Q})\), with each \(x_iH_i\) contained in \(X\). This \(\varepsilon\) depends only on \(N\) and the degree of \(X\). In a similar spirit, it is known that \(X^0\cap C(\overline\Gamma,\varepsilon)\) is finite for some \(\epsilon>0\), where \(x\in X^0\) if \(x\in X\) but \(xH\not\subset X\) for any positive-dimensional subgroup \(H\) of \({\mathbb G}_m^N(\overline{\mathbb Q})\).
There are diophantine approximation results explicitly bounding heights of points in \(X\cap\overline\Gamma\) (etc.) in certain cases – for instance, for most curves \(X\) in the case \(N=2\) – coming from Baker-type logarithmic forms estimates. Using these results and some estimates of the number of points of small heights, the authors make the descriptions above effective, for certain classes of \(X\). For \(X\cap \overline\Gamma_\varepsilon\), they give an explicit \(\varepsilon\) and effectively computable bounds on the heights and degrees of the points \(x_i\). For \(X\cap C(\overline\Gamma,\varepsilon)\) they again give an explicit value for \(\varepsilon\) and also bounds on the heights and degrees of the elements of \(X\cap C(\overline\Gamma,\varepsilon)\).
Although the class of varieties \(X\) considered is rather restricted (either \(N=2\) and \(X\) a curve, or \(X\) cut out by bi- and trinomials), the method is more generally applicable at least in principle.