Let k be a number field, let θ be a nonzero algebraic number, and
let H(â‹…) 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∈k with H(αθ)≤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.Comment: accepted for publication by Ann. Sc. Norm. Super. Pisa Cl. Sci., 201