# Fine tuning the growth rate of the degrees of polynomials

Let $$r$$ be an integer with $$r>1$$. Suppose that if $$k\geq 0$$, then $$p_{k}(x)$$ is a polynomial with nonnegative integer coefficients with $$p_{k}(0)=1$$ but where $$p_{k}\neq 1$$.

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

For each $$n>0$$, let $$t_{n}$$ be the number indices $$k$$ where $$\deg(p_{k}(x))=n$$. Then is it possible to select polynomials $$(p_{k})_{k\geq 0}$$ where $$|t_{n}-\frac{r^{n}}{n}|=O(\alpha^{n})$$ for each $$\alpha>1$$?

How slowly can the function $$n\mapsto|t_{n}-\frac{r^{n}}{n}|$$ grow? How slowly can the function $$n\mapsto\max(0,\frac{r^{n}}{n}-t_{n})$$ grow? For example, can we have $$\max(0,\frac{r^{n}}{n}-t_{n})=O(\alpha^{n})$$ for all $$\alpha>1$$?

This question is motivated by very large cardinals.

• It's not so hard to get $\alpha = \sqrt{r}$, or, for the second version, $r^{1/3}$. – Will Sawin Feb 14 at 3:57