Time complexity of multidimensional Turing machines

It is proved that the work of an indeterminate m-dimensional Turing machine with time complexity t can be simulated on an indeterminate k-dimensional (k≤m) Turing machine with time complexity t1−(1/m)+(1/k)+ɛ (for any ε>0). Moreover, the following generalization to the multidimensional case of the familiar theorem of Hopcroft, Paul, and Valiant is proved: the work of an m-dimensional Turing machine with time complexity t log1/mt [t(n)≥n] can be simulated on an address machine working with time complexity t.