**Intro**

The question is about Game of Life.

Let us denote the set of points obtained from initial configuration $A$ after $m$ steps as $A(m)$ (we are only interested in finite initial configuration, i.e. those one which formed by finite number of marked cells).

Let us denote the number of marked cells at configuration $A$ as $N(A)$. Then increment at $m$-th step could be computed as $i(A, m) = \max(0, N(A(m+1)) - N(A(m))$.

Majority of configurations doesn't grow in size after some number of steps. There are known examples which grows linearly or quadratically in time. For all of this examples the relative increment decay as times goes by: $\frac{i(A,m)}{N(A(m))} \to 0$ as $m \to \infty$.

**Question**

Is it possible to prove some uniform result of this type:

is it true that $\forall \epsilon > 0$ (may be some other restriction) $\exists M_\epsilon$ such that $\forall A$ (for any arbitrary chosen initial configuration) and $\forall m > M_\epsilon$: $\frac{i(A,m)}{N(A(m))} < \epsilon$?

May be it makes sense for some good family of initial configurations $A$?

This is rather a probe question which I asked with a hope to find some references or ideas in answers which will direct me in more useful settings.