Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2 x 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 Z[1/root 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 Z[1/root 2, i]. We focus on the subrings Z[1./2], Z[1/root 2], Z[1/i root 2], and 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 Hadarnard gate and an optional phase gate.