The original problem is following:
Problem 1 $\lim_{N\to \infty}|\frac{\#\{1\leq n\leq N |\ \ \ [n\sqrt2]=0(mod 2)\}}{N}|=\frac{N}{2}+O(ln(N))$.
This problem is not very difficult, in fact $\#{\{1\leq n\leq N |\ \ \ [n\sqrt2]=0(mod 2)\}}=\#\{1\leq n\leq N| \{n\frac{\sqrt 2}{2}\}\in (0,\frac{1}{2})\} $.
and due to the observation $\frac{\sqrt 2}{2}$ is smooth and following effective uniformly distribution result:
Theorem (effective uniformly distribution for smooth irrational number) $\alpha \in \mathbb R- \mathbb Q$ is a smooth irrational number, then we have, $\forall 0<a<b<1$, $$\#\{1\leq n\leq N\ |\ \{\alpha n\}\in (a,b) \}=(b-a)N+O(log(N))$$
The proof of the theorem is a careful look at the continue fractional of $\alpha$. Now we shift to two natural emerge problems, but more difficult. I do not know how to handle them.
Problem 2 if $\alpha$ is an irrational number. $P(n)$ is a polynomial with integer coefficient, under what assume on $\alpha$ we have, $$\#\{1\leq n\leq N\ |\ \{\alpha P(n)\}\in (a,b) \}=(b-a)N+O(log(N))$$ Is it enough if we assume $\alpha$ is smooth?
I do not know if the problem is a good question, may be we should change the error term $log(N)$ to something else.
A remark, if we just care about the uniformly distribution, this could be done by Weyl method or van der coput trick, the key point is what can we say about the error term as better as possible?
Problem 3 if $\alpha$ is an irrational number. $Q(n)$ is a polynomial with integer coefficient, under what assume on $\alpha$ we have good asymptotic for, $$\#\{1\leq n\leq N\ |\ \exists m\in \mathbb N^*,[\alpha n]=Q(m) \}?$$ Is it enough if we assume $\alpha$ is smooth to gain such a asymptotic?