Consider a number $x$ with known irrationality measure $r$ (for example $\pi$ with $7<r<8$). Is there anything we can say about the irrationality measure of a polynomial \begin{align} p=p(x)=\sum_{n\in N}c_nx^n \end{align} where $N\subset\mathbb{N}$ is some finite index set and the coefficients $c_n$ are rational (or even $c_n=1$ for all $n$, if that makes things easier)? Will it be finite? Does it depend on $N$?
I don't know much about number theory, any help would be much appreciated!
Let's assume the coefficients are integers. If $|x - p/q| < c/q^\alpha$, then $|f(x) - f(p/q)| \le Kc |x - p/q| < Kc/q^\alpha$ where $K$ is an upper bound for $|f'|$ in a neighbourhood of $x$. Since $f(p/q)$ has denominator $\le q^d$ where $f$ has degree $d$, we find that the irrationality measure of $f(x)$ is at least $r/d$.