given any total recursive function g(n) ≥ n
there exists a total recursive function S(n) such that DTIME(S(n)) = DTIME(g(S(n)))
(see complexity classes). In other words there is a “gap” between time bounds S(n) and g(S(n)) within which the minimal space complexity of no language lies.
This has the following counter-intuitive consequence: given two universal models of computation, say a Turing machine that makes one move per century and the other a random-access machine capable of performing a million arithmetic operations per second, then there is a total recursive function S(n) such that any language recognizable in time S(n) on one machine is also recognizable within time S(n) on the other.
"gap theorem." A Dictionary of Computing. . Encyclopedia.com. (December 13, 2018). https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/gap-theorem
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