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Schanuel's theorem for heights defined via extension fields

Let $k$ be a number field, let $θ$ be a nonzero algebraic number, and let $H(\cdot)$ be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of $α\in k$ with $H(αθ)\leq X$. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of $k$. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof. Moreover, we analyze the leading constant in our asymptotic formula for Loher and Masser's problem. In particular, we prove a sharp upper bound in terms of the classical Schanuel constant.