  In this letter, we discuss certain universal predictions of the large charge
expansion in conformal field theories with $U(1)$ symmetry, mainly focusing on
four-dimensional theories. We show that, while in three dimensions quantum
fluctuations are responsible for the existence of a theory-independent $Q^0$
term in the scaling dimension $\Delta_Q$ of the lightest operator with fixed
charge $Q\gg 1$, in four dimensions the same mechanism provides a universal
$Q^0\log Q$ correction to $\Delta_Q$. Previous works discussing
four-dimensional theories failed in identifying this term. We also compute the
first subleading correction to the OPE coefficient corresponding to the
insertion of an arbitrary primary operator with small charge $q\ll Q$ in
between the minimal energy states with charge $Q$ and $Q+q$, both in three and
four dimensions. This contribution does not depend on the operator insertion
and, similarly to the quantum effects in $\Delta_Q$, in four dimensions it
scales logarithmically with $Q$.
