Capacity equalities in 1-dimensional (d, k)-constrained systems

In this paper, we consider the problem of determining when the capacities of distinct 1-dimensional (d, k)-constrained systems can be equal. If we let C(d, k) denote the capacity of a (d, k)-constrained system, then it is known that C(d, 2d) = C(d + 1, 3d + 1), and C(d, 2d + 1) = C(d + 1, infinity). Repeated application of these two identities also yields the chain of equalities C(1, 2) = C(2,4) = C(3,7) = C(4, infinity). We show that these are the only equalities possible among the capacities of (d, k)-constrained systems.