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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

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    $\begingroup$ Please use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$
    – GH from MO
    Commented Sep 1, 2022 at 21:40
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    $\begingroup$ If $\alpha=1$, then the fractional part of $\alpha n^2/q$ is $j/q$ where $j$ is a quadratic residue modulo $q$, so lots of intervals of length $1/q$ will be empty. $\endgroup$
    – Gerry Myerson
    Commented Sep 2, 2022 at 0:51
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    $\begingroup$ Still doesn't make sense, if $\alpha$ is super tiny with respect to $q$, say $q^{-(2+\epsilon}$, you will have only one $i$ which is non-zero (with high multiplicity). Obviously for a reasonable $\alpha$ (with respect to the range) one can get an estimate through Gaussian sums, as Gerry alluded to. $\endgroup$
    – Asaf
    Commented Sep 2, 2022 at 16:02
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    $\begingroup$ I don't know whether this helps, or is even closely related, but it sort of looks like the kind of thing you're after: by Theorem 3.2 of Kuipers and Niederreiter, Uniform Distribution of Sequences, for $\alpha$ a real irrational the sequence $\alpha n^2$, $n=1,2,\dots$, is uniformly distributed modulo one. That book is in any event a good place to start any search for results on distribution of sequences. $\endgroup$
    – Gerry Myerson
    Commented Sep 3, 2022 at 1:39
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    $\begingroup$ @JohnnyT. for an appropriate $\alpha$, this is sum over quadratic residues. One may expand the relevant counting function associated to $i/q$ by the exponentials, and then analyze the related sums. $\endgroup$
    – Asaf
    Commented Sep 4, 2022 at 17:19

1 Answer 1

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Lemma 1 of

http://www.numdam.org/item/ASNSP_1999_4_28_4_591_0.pdf

bounds $\{ d\sim D\text { with }||\alpha d^2||<D\} $ and gives a non-trivial result for $D\approx q$... not sure if this is enough to help you with the $i$ conditions, but maybe it's worth taking a look. If I'm overlooking something stupid I'll delete.

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  • $\begingroup$ i have a look. thank you $\endgroup$
    – Johnny T.
    Commented Sep 5, 2022 at 14:29

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