Multiplicative approximation by the Weil height

Abstract:

Let $K/\mathbb {Q}$ be an algebraic extension of fields, and let $\alpha \not = 0$ be contained in an algebraic closure of $K$. If $\alpha$ can be approximated by roots of numbers in $K^{\times }$ with respect to the Weil height, we prove that some nonzero integer power of $\alpha$ must belong to $K^{\times }$. More generally, let $K_1, K_2, \dots , K_N$, be algebraic extensions of $\mathbb {Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $\alpha \not = 0$ can be approximated by a product of roots of numbers from each $K_n$ with respect to the Weil height, we prove that some nonzero integer power of $\alpha$ must belong to the multiplicative group $K_1^{\times } K_2^{\times } \cdots K_N^{\times }$. Our proof of the more general result uses methods from functional analysis.