Let $\Gamma \subset \overline {\mathbb {Q}}^*$ be a finitely generated subgroup. Denote by $\Gamma _{\mathrm {div}}$ its division group. A recent conjecture due to Rémond, related to the Zilber–Pink conjecture, predicts that the absolute logarithmic Weil height of an element of $\mathbb {Q}(\Gamma _{\mathrm {div}})^*\backslash \Gamma _{\mathrm {div}}$ is bounded from below by a positive constant depending only on $\Gamma $. In this paper, we propose a new way to tackle this problem.

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $-function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$, respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points.

We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$ , which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$ .

For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$, let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$. We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$. For each fixed $\unicode[STIX]{x1D709}>0$, these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$, which was conjectured in a question at Mathoverflow [https://mathoverflow.net/questions/177206/].

Let 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to a family of double complex integrals, we prove ℚ(α)-irrationality measures and ℚ(α)-nonquadraticity measures of log α for several algebraic numbers α∈ℂ, improving earlier results due to Amoroso and the second-named author.

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha $ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.