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Simultaneous Diophantine approximation is concerned with the approximation ofa point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with aview towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and$H(\mathbf r)$. Here $H(\mathbf r)$ is the so-called "standard height" of therational point $\mathbf r$. In this paper the authors ask: What changes if wereplace the standard height function by a different one? As it turns out, thischange leads to dramatic differences from the classical theory and requires thedevelopment of new methods. We discuss three examples of nonstandard heightfunctions, computing their exponents of irrationality as well as giving moreprecise results. A list of open questions is also given.

Unconventional height functions in simultaneous Diophantine approximation