# Distribution of $\alpha n^2/q$ modulo $1$?

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

• Please use a high-level tag like "nt.number-theory". I added this tag now. Sep 1, 2022 at 21:40
• 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. Sep 2, 2022 at 0:51
• 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.
– Asaf
Sep 2, 2022 at 16:02
• 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. Sep 3, 2022 at 1:39
• @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.
– Asaf
Sep 4, 2022 at 17:19

bounds $$\{ d\sim D\text { with }||\alpha d^2|| and gives a non-trivial result for $$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.