Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$. I am interested in upper bound for $$ M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \} $$ where $N$ is a sufficiently large positive integer with respect to $I$. I was wondering is it reasonable to expect $$ \sup_{\alpha \in I} M(\alpha) \ll N^{\varepsilon} ? $$ or would one expect this to be not true? (or rather the statement holds for $\sup_{\alpha \in I - B}$ and some subset $B$. Is it possible to bound the measure of $B$?).

The question is related to my question from couple years ago (Distribution of $\alpha n^2/q$ modulo $1$?) in which a reference to Lemma 1 of http://www.numdam.org/item/ASNSP_1999_4_28_4_591_0.pdf was given. Here an upper bound is given, but it wasn't clear to me if it was sharp or not. Any input would be appreciated. Thank you.

Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$. I am interested in upper bound for $$ M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \} $$ where $N$ is a sufficiently large positive integer with respect to $I$. I was wondering is it reasonable to expect $$ \sup_{\alpha \in I} M(\alpha) \ll N^{\varepsilon} ? $$ or would one expect this to be not true? (or rather the statement holds for $\sup_{\alpha \in I - B_N}$ and some subset $B_N$ that depends on $N$. Is it possible to bound the measure of $B_N$ in terms of $N$?).

The question is related to my question from couple years ago (Distribution of $\alpha n^2/q$ modulo $1$?) in which a reference to Lemma 1 of http://www.numdam.org/item/ASNSP_1999_4_28_4_591_0.pdf was given. Here an upper bound is given, but it wasn't clear to me if it was sharp or not. Any input would be appreciated. Thank you.

Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$. I am interested in upper bound for $$ M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \} $$ where $N$ is a sufficiently large positive integer with respect to $I$. I was wondering is it reasonable to expect $$ \sup_{\alpha \in I} M(\alpha) \ll N^{\varepsilon} ? $$ or would one expect this to be not true?

The question is related to my question from couple years ago (Distribution of $\alpha n^2/q$ modulo $1$?) in which a reference to Lemma 1 of http://www.numdam.org/item/ASNSP_1999_4_28_4_591_0.pdf was given. Here an upper bound is given, but it wasn't clear to me if it was sharp or not. Any input would be appreciated. Thank you.

Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$. I am interested in upper bound for $$ M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \} $$ where $N$ is a sufficiently large positive integer with respect to $I$. I was wondering is it reasonable to expect $$ \sup_{\alpha \in I} M(\alpha) \ll N^{\varepsilon} ? $$ or would one expect this to be not true? (or rather the statement holds for $\sup_{\alpha \in I - B}$ and some subset $B$. Is it possible to bound the measure of $B$?).

The question is related to my question from couple years ago (Distribution of $\alpha n^2/q$ modulo $1$?) in which a reference to Lemma 1 of http://www.numdam.org/item/ASNSP_1999_4_28_4_591_0.pdf was given. Here an upper bound is given, but it wasn't clear to me if it was sharp or not. Any input would be appreciated. Thank you.