Simulations among multidimensional turing machines

For all d ≥ 1 and all e ≫ d, every deterministic multihead e-dimensional Turing machine of time complexity T(n) can be simulated on-line by a deterministic multihead d-dimensional Turing machine in time O(T(n)1+1/d-1/e(log T(n))O(1)). This simulation almost achieves the known lower bound Ω(T(n)1+1/d-1/e) on the time required. Furthermore, there is a deterministic d-dimensional machine with just two worktape heads that simulates the e-dimensional machine on-line in time O(T(n)1+1/d-1/delog T(n)). These simulations are interpreted in terms of dynamic embeddings among data structures.