On \(U_m\)-numbers with small transcendence measure (Q1878570)

Let \(x\) be a complex number. Assume that \(w^*_n(x)\) is the supremum of the real numbers \(r\) for which there exist infinitely many algebraic numbers \(a\) of degree at most \(n\) satisfying \[ 0<| x-a| <H(a)^{-r-1}, \] where \(H(a)\) denotes the maximum of the absolute values of the coefficients of its minimal defining polynomial of the number \(a\) over \(\mathbb{Z}\). The main result of this paper is that if \(m>1\), then there exist infinitely many \(U_n\)-numbers \(x\) with \[ w^*_n(x) \leq m+n-1, \] for every \(1\leq n<m\).