Fix a large positive integer $m$. Let $A$ be small positive number typically $\sim m^{1/3}$. Suppose $S(A, m)$ be set of solutions (normalized by dividing by $m$) to the quadratic congruences $x^2 = a \mod{m}$ such that $0 < x < m$ as $a$ runs from $1$ to $A$. What is known about the distribution of $S(A, m)$ on $[0, 1)$? Does it show any form of equidistribution? If yes then what error terms does it give?
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$\begingroup$ You mean that $A$ is a set of quadratic residues? $\endgroup$– AsafCommented May 8, 2023 at 22:49
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$\begingroup$ @Asaf $A$ is a number. There could be both quadratic residues and non residues between $1$ and $A$. $\endgroup$– MelankaCommented May 8, 2023 at 22:52
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2$\begingroup$ If $A$ is slightly larger than $m^{1/2}$, then $S(A,m)$ has roughly the "expected" size, which I believe is about $A/2^{\omega(m)}$ (give or take a factor if $m$ is even), by Pólya-Vinogradov-type estimates. We of course don't expect such uniformity in smaller ranges; whether $m^{1/3}$ is enough for some form of equidistribution seems like a fairly subtle question to me $\endgroup$– WojowuCommented May 9, 2023 at 0:36
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$\begingroup$ @Wojowu: do you mind sketching a bit more what you meant in your comment? I spent a little while thinking about this problem, and I am not sure how you are applying Pólya--Vinogradov. $\endgroup$– Anurag SahayCommented Nov 18, 2023 at 17:54
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