# Asymptotics – fine-tuning the growth rate of polynomial degrees

To let $$r$$ be an integer with $$r> 1$$, Suppose that $$p_ {k} (x)$$ is a polynomial with positive integer coefficients with $$p_ {k} (0) = 1$$ but where $$p_ {k} neq 1$$ for all $$k geq 0$$,

Suppose that
$$prod_ {k = 0} ^ { infty} p_ {k} (x) = frac {1} {1-rx}$$

For each $$n> 0$$, To let $$t_ {n}$$ be the number indexes $$k$$ from where $$deg (p_ {k} (x)) = n$$, Then polynomials can be selected $$(p_ {k}) _ {k geq 0}$$ from where
$$| t_ {n} – frac {r ^ {n}} {n} | = O ( alpha ^ {n})$$
for each $$alpha> 1$$?

How slow can this work? $$n mapsto | t_ {n} – frac {r ^ {n}} {n} |$$ to grow? How slow can this work? $$n mapsto max (0, frac {r ^ {n}} {n} -t_ {n})$$ to grow? For example we can have $$max (0, frac {r ^ {n}} {n} -t_ {n}) = O ( alpha ^ {n})$$ for all $$alpha> 1$$?

This question is motivated by very great cardinals.