Let $Q=\{q_n\}_{n=1}^{\infty}$ be a sequence of integers greater than or
equal to $2$. We say that a real number $x$ in $[0,1)$ is {\it $Q$-distribution
normal} if the sequence $\{q_1q_2 \... q_n x\}_{n=1}^{\infty}$ is uniformly
distributed mod $1$. In \cite{Laffer}, P. Laffer asked for a construction of a
$Q$-distribution normal number for an arbitrary $Q$. Under a mild condition on
$Q$, we construct a set $\Theta_Q$ of $Q$-distribution normal numbers. This set
is perfect and nowhere dense. Additionally, given any $\alpha$ in $[0,1]$, we
provide an explicit example of a sequence $Q$ such that the Hausdorff dimension
of $\Theta_Q$ is equal to $\alpha$. Under a certain growth condition on $q_n$,
we provide a discrepancy estimate that holds for every $x$ in $\Theta_Q$.