# H4

There exist a fixed integer \(m\ge1\), a fixed real number \(\gamma>0\), a sequence of number fields \((K_j)_{j\ge1}\), and finite sets \(U_j\subset K_j\), such that, writing \(d_j=[K_j:\mathbb Q]\), one has
\[
d_j\longrightarrow\infty,
\qquad
U_j\subset m^{-1}\mathcal O_{K_j},
\qquad
|U_j|\ge\exp(\gamma d_j).
\]
For every \(j\ge1\), every \(u\in U_j\), and every field embedding \(\sigma:K_j\hookrightarrow\mathbb C\),
\[
|\sigma(u)|=1.
\]

Here \(\mathcal O_{K_j}\) is the ring of integers of \(K_j\), and \(m^{-1}\mathcal O_{K_j}=\{a/m:a\in\mathcal O_{K_j}\}\). The same constants \(m\) and \(\gamma\) apply to every \(j\).