2
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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?

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  • $\begingroup$ An irrational number is said to be smooth if the terms of its continued fraction are bounded, correct? For instance $\sqrt{2}$ is smooth since $\sqrt{2}=[1;\overline{2}]$ and $\varphi$ is smooth since $\varphi=[1;\overline{1}]$? $\endgroup$ – Jack D'Aurizio Dec 10 '17 at 13:23
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    $\begingroup$ @JackD'Aurizio, yeah you are right, exactly. $\endgroup$ – Hu xiyu Dec 10 '17 at 13:24
  • $\begingroup$ The statement of problem $2$ should be simple to prove if $O(\log n)$ is replaced by $O\left[\left(\log n\right)^{\deg P}\right]$ or something like that, by just mimicking the proof of the first Theorem, i.e. by considering suitable exponential sums and applying Weyl's bound. $\endgroup$ – Jack D'Aurizio Dec 10 '17 at 13:27
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    $\begingroup$ @JackD'Aurizio, good point! I have not realized this truth until you point out! thanks a lots! $\endgroup$ – Hu xiyu Dec 10 '17 at 13:29
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    $\begingroup$ Landau's conjecture alone is a horribly difficult problem, but your Problem $3$ is much different in my opinion. $\lfloor \alpha n\rfloor$ form a Beatty sequence with a positive density, so it is very likely that the range of almost any polynomial with integer coefficients intersects such sequence infinite times. $\endgroup$ – Jack D'Aurizio Dec 10 '17 at 13:43

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