Abstract
Let k be a number field, let ✓ be a nonzero algebraic number, and let H(·) 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 ↵ 2 k with H(↵✓) X, and we analyze the leading constant in our asymptotic formula. In particular, we prove a sharp upper bound in terms of the classical Schanuel constant. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of k with a fairly explicit error term. This provides a conceptual framework for Loher and Masser’s problem and generalizations thereof. Finally, we establish asymptotic counting results for varying ✓, namely, for the number of pp↵ of bounded height, where ↵ 2 k and p is any rational prime inert in k.
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