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Let $\alpha \in \mathbb{R} \backslash \mathbb{Q}$ and $q$ a positive integer. Let $||.||$ denote the distance to the closest integer and define $$ N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \alpha n^2/q || < \frac{i+1}{q}}} 1 $$ for each $0 \leq i < q/2$. Is it possible to show that for $q$ sufficiently large $N_i(q) > 0$ holds for all, or for 'most', $i$ in consideration? or are there $\alpha$ for which something like this can be proved?

I first thought Weyl criterion could be useful but I couldn't manage to make it work... Any inputs appreciated. thank you!

Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer. Let $||.||$ denote the distance to the closest integer and define $$ N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \alpha n^2/q || < \frac{i+1}{q}}} 1 $$ for each $0 \leq i < q/2$. I would like to obtain an upper bound for $$ \max_{0 \leq i < q/2} N_i(q) \leq f(q) $$ $f$ is some function of $q$ that is not the trivial bound. Is it possible to prove something like this? where the bound is uniform in $\alpha$? I'm curious to know what is available so any reference to related topic is appreciated as well. thank you!

edit. question was changed a lot. initially I asked for most $N_i(q)$ to be non zero

Let $0 \neq \alpha \in \mathbb{R}$ and $q$ a positive integer. Let $||.||$ denote the distance to the closest integer and define $$ N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \alpha n^2/q || < \frac{i+1}{q}}} 1 $$ for each $0 \leq i < q/2$. Is it possible to show that for $q$ sufficiently large $N_i(q) > 0$ holds for all, or for 'most', $i$ in consideration? I first throught Weyl criterion could be useful but I couldn't manage to make it work... Any inputs appreciated. thank you!

Let $\alpha \in \mathbb{R} \backslash \mathbb{Q}$ and $q$ a positive integer. Let $||.||$ denote the distance to the closest integer and define $$ N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \alpha n^2/q || < \frac{i+1}{q}}} 1 $$ for each $0 \leq i < q/2$. Is it possible to show that for $q$ sufficiently large $N_i(q) > 0$ holds for all, or for 'most', $i$ in consideration? or are there $\alpha$ for which something like this can be proved? 

I first thought Weyl criterion could be useful but I couldn't manage to make it work... Any inputs appreciated. thank you!