>**Problem 1**
Given a irrational number $\alpha$ and two polynomials with positive integer coefficients $P(n),Q(n)$, is it possible to get the asymptotic estimate and reasonable error term for:
$$\lim_{N\to \infty}\#\{1\leq n\leq N \ | \ \exists m\in \mathbb N,  [P(n)\alpha]=Q(m)\}?$$

In the case when $Q(n)=an+b$, by Weyl method or van der curput trick it is not difficult to establish such a estimate at least for the asymptotic part, and if we know $\alpha$ is a smooth irrational number, we could even say some thing about the error term. In fact for smooth irrational number we have the following "effective uniformly distribution result".

>**Theorem (effective uniformly distribution)**
$\alpha \in \mathbb R- \mathbb Q$ is a smooth irrational number, then we have, $\forall 0<a<b<1$,
$$\#\{1\leq n\leq N\ |\ \{\alpha P(n)\}\in (a,b) \}=(b-a)N+O(log^{deg(P)}(N))$$

The proof of this theorem is based on a careful look at the continual fraction expansion of $\alpha$.

In the case $P(n)=n$ and $Q(n) $ is a arbitrarily monic integer polynomial (the highest order coefficient is 1) and $\alpha$ is smooth this problem is also not very difficult, we could establish the following asymptotic expansion,

$$\lim_{N\to \infty}\#\{1\leq n\leq N\ |\ \exists m\in \mathbb N^*,\{n\alpha \}=Q(m) \}=N^{\frac{1}{deg(Q)}}$$

Although it seems not very easy to get a reasonable order of error term.

Although the general situation may be complicated, the typical situation I wondering to understand is the case $Q(n)=n^k+1$, and $P(n)$ is a arbitrary polynomial with positive integer coeficients, I conjecture the following asymptotic is true.
>**Problem 2**
$\alpha$ is a irrational number, $P(n)$ is a polynomial with integer coefficient, could the following asymptotic formula be true?
$$\lim_{N\to \infty}\frac{\#\{1\leq n\leq N\ |\ \exists m\in \mathbb N,[P(n)\alpha ]=m^k+1 \}}{N^{\frac{1}{k}}}=1$$

I will appreciate for any valuable comments and advice.