  Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\times 2$ unitary
matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if
and only if the entries of $V$ belong to the ring $\mathbb{Z}[1/\sqrt{2},i]$.
Later that year, Giles and Selinger showed that the same restriction applies to
matrices that can be exactly represented by a multi-qubit Clifford+$T$ circuit.
These number-theoretic characterizations shed new light upon the structure of
Clifford+$T$ circuits and led to remarkable developments in the field of
quantum compiling. In the present paper, we provide number-theoretic
characterizations for certain restricted Clifford+$T$ circuits by considering
unitary matrices over subrings of $\mathbb{Z}[1/\sqrt{2},i]$. We focus on the
subrings $\mathbb{Z}[1/2]$, $\mathbb{Z}[1/\sqrt{2}]$,
$\mathbb{Z}[1/i\sqrt{2}]$, and $\mathbb{Z}[1/2,i]$, and we prove that unitary
matrices with entries in these rings correspond to circuits over well-known
universal gate sets. In each case, the desired gate set is obtained by
extending the set of classical reversible gates $\{X, CX, CCX\}$ with an
analogue of the Hadamard gate and an optional phase gate.
