  It is well known that all numbers that are normal of order $k$ in base $b$
are also normal of all orders less than $k$. Another basic fact is that every
real number is normal in base $b$ if and only if it is simply normal in base
$b^k$ for all $k$. This may be interpreted to mean that a number is normal in
base $b$ if and only if all blocks of digits occur with the desired relative
frequency along every infinite arithmetic progression. We reinterpret these
theorems for the $Q$-Cantor series expansions and show that they are no longer
true in a particularly strong way. The main theoretical result of this paper
will be to reduce the problem of constructing normal numbers with certain
pathological properties to the problem of solving a system of Diophantine
relations.
