Given a deterministic Turing machine T which begins on an infinite blank strip, let its growth rate $G_T(t)$ represent the number of non-blank squares after the machine is run for $t$ time steps.
It is an amusing puzzle to try to construct a machine with given growth rate asymptotics. A linear growth rate is very easy to achieve (simply have the machine constantly move right and write a $1$ every step). Here is an example of a machine with $G_T(t) = \Theta(\sqrt{t})$.
A natural question is, which Turing machines have the fastest asymptotic growth rates? Clearly, linear wins (as for all Turing machines $T$, $G_T(t) \le t$). But what about second place? What is the fastest possible sublinear growth rate of a Turing machine?
In particular, I have the following conjecture: For all Turing machines $T$, $$G_T(t)=\omega\left({t\over \log(t)}\right) \implies G_T(t)=\Theta(t)$$
In words, the growth rate of any Turing machine which asymptotically dominates $t\over \log(t)$ must be asymptotically linear. (So, for example, the conjecture would imply there cannot be a Turing machine with growth rate asymptotic to $t/\sqrt{\log t})$.
My conjecture is motivated by the this second-place candidate I constructed (coupled with a rather nonrigorous argument for its asymptotic growth rate). It works rather simply: it repeatedly increments a binary string, moves it left, and leaves a trail of $1$'s behind. The time it takes to increment and move the string is linear in the length of the binary string, hence the leftward 'velocity' scales with the inverse of the length of this string. The length of the string grows logarithmically over time, hence the leftward velocity is asymptotic to $1\over \log(t)$. Thus the overall distance travelled with time is asymptotic to $\int{1\over \log t} dt = \Theta({t\over \log t})$. The trail of $1$'s will dominate the population in the long term, and its length is approximately the distance the binary string has travelled, hence the growth rate of this machine is asymptotically $\Theta({t\over \log t})$.
I am unable to conceive of a faster sublinear machine, hence the above conjecture. Is this conjecture correct? Or can you find a machine whose growth rate is sublinear yet dominates mine?