Unconventional height functions in simultaneous Diophantine approximation

Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf{x}\in {\mathbb{R}}^d$ by points $\mathbf{r}\in {\mathbb{Q}}^d$, with a view towards jointly minimizing the quantities $\mathrm{\parallel }\mathbf{x}-\mathbf{r}\mathrm{\parallel }$ and $H(\mathbf{r})$. Here $H(\mathbf{r})$ is the so-called “standard height” of the rational point $\mathbf{r}$. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.