I ask this question on math stackexchange, but there is no answer, so please forgive me I ask it here again.
Let $c>0$ and $P(x)$ be a polynomial. Then there exists a $p>1$ (e.g. we can take $p$ to be greater than the degree of $P$) such that for almost all $x=e^{2\pi i\theta},\theta\in[0,1]$ on the unit circle, $∣P(x^N)∣<\frac{c}{N^p}$ occurs only for finitely many $N\in \mathbb{N}$.
We can show this for $P(x)=x−1$. $|P(x^N)|=|x^N−1|=|e^{2\pi iN\theta}−1|=|e^{2\pi i(N\theta−m)}−1|∼|N\theta−m|$ where $m$ is the closest integer to $N\theta$. For any $p>1$, applying this result proved by Borel-Cantelli Lemma gives the result.
For general $P(x)$, we only need to factorize it over $\mathbb{C}$ and analyse the factors similarly.
I believe similar result also holds for multivariate polynomial $P(x_1,x_2,\dots,x_n)$. Namely, there exists a $p>1$ such that for almost all $x_1=e^{2\pi i\theta_1},\dots,x_n=e^{2\pi i\theta_n}$ on the $n$-dimentional torus, $∣P(x_1^N,\dots,x_n^N)∣<\frac{c}{N^p}$ occurs only for finitely many $N\in \mathbb{N}$.
Moreover I guess this holds for several variable analytic functions.
Could you prove this or give a reference please? Thanks.
