Let $n$ be a positive integer and $0 \leq i < n$. Define $$ N(i) = \# \left\{ (x_1,\dots, x_s) \in [1, n]^s: x_1^2 +\dots + x_s^2 \equiv i \mod n \right\}. $$ I am looking for a reference for the following result: if $s$ is large enough, $$ n^{s1} \ll N(i) \ll n^{s1} $$ is true for all $i$ and $n$. Could anyone provide a reference for this result? Thank you! Any input also appreciated.

$\begingroup$ surely you mean the size of the set is $N(i)$ $\endgroup$– kodluSep 9 at 7:46

$\begingroup$ yes, thank you. fixed $\endgroup$– SJYSep 9 at 7:52

2$\begingroup$ For fixed $n$, you can think about this as a length$s$ random walk on the directed graph with vertex set $\mathbb Z/n\mathbb Z$ and edges corresponding to addition of squares. As usual, this converges to the stationary distribution with error exponentially small in $s$, that is, $n^{1}n^{s}N(i)=\exp(\Omega(s))$ (where the implied constants in the $\Omega$ notation depend on $n$). $\endgroup$– Emil JeřábekSep 9 at 8:41
2 Answers
The estimate is false for $s\leq 4$, but it is true for $s\geq 5$. For example, if $s=3$ and $8\mid n$ and $i=7$, then $N(i)=0$. Or if $s=4$ and $n=4^k$ and $i=0$, then $N(i)\leq n^2$. For $s\geq 5$ the result follows easily from the fact that the number of ways to write a positive integer $m$ as a sum of $s$ squares is $\asymp_s m^{s/21}$. For more details, see Vaughan's book "The HardyLittlewood method".
Added. Here are some additional details. From the mentioned bound $$r_s(m):=\#\{ x_1,\ldots,x_s \in \mathbb{Z}: x_1^2+\ldots + x_s^2 = m\} \asymp_s m^{s/2  1},$$ it follows that \begin{align*} N(i)&\leq\sum_{k=1}^{sn+1} r_s(knn+i)\ll_s \sum_{k=1}^{sn+1}(kn)^{s/21}\ll_s n^{s1},\\ N(i)&\geq\sum_{k=1}^{n} r_s(knn+i)\gg_s \sum_{k=1}^{n}(kn)^{s/21}\gg_s n^{s1}. \end{align*}

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$\begingroup$ @GHfromMO Can you explain, in the comments or in the answer, how do you conclude that $N(i)\asymp_s n^{s1}$ from $r_s(m):=\#\{ x_1,\ldots,x_s \in \mathbb{Z}: x_1^2+\ldots + x_s^2 = m\} \asymp_s m^{s/2  1}$? I don't see a quick way to do it but it does sound intuitive. (What confuses me is that as far as I can tell, the squares appearing in $r_s(m)$ do not necessarily behave like random squares in $\mathbb{Z}/n\mathbb{Z}$) $\endgroup$ Sep 10 at 13:24

2$\begingroup$ Thank you for your answer! $N(i)$ is the same as the number of $(x_1, ..., x_n) \in [1, n]^s$ with $x_1^2 + ... + x_s^2$ equal to one of $i$, $n+i$, ..., $(sn1)n + i$. so we can use the mentioned estimate for each of these and add. This was my understanding of @GHfromMO's comment $\endgroup$– SJYSep 10 at 14:54

1$\begingroup$ @OfirGorodetsky As SJY said, you simply need to add up the bounds for $r_s(m)$. See my "Added" section for more details. $\endgroup$ Sep 10 at 15:32

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For simplicity, I will focus on odd $n$ only. I will prove the following estimate:
Claim: Uniformly for $s \ge 5$ and odd $n$ we have $N(i) = n^{s1} (1+O(p^{1\frac{s}{2}}))$ where $p$ is the smallest prime dividing $n$, and this is optimal if $i=0$ and $4\mid s$. We also have $N(i) \gg n^{s1}$, uniformly in the same range. All implied constants are absolute.
But first, I want to elaborate on GH from MO's answer and Emil Jeřábek's proof in the comments.
Circle method: If one wants to count integer solutions to $x_1^2+ \ldots+x_s^2 = n$ (a special case of Waring's problem) then one can use the circle method, which for large enough $s$ relates this problem to the local problem of counting solutions to $x_1^2+\ldots+x_n^2 \equiv i \bmod n$ where $x_i$ are variables modulo $n$. So people have studied your $N(i)$ in this context (among other contexts). As GH from MO, in this case one can recover information about the local problem (estimating $N(i)$) from the global one.
Quadratic forms: You are counting solutions to $Q(x_1,\ldots,x_s)=i \bmod n$ for a (diagonal) quadratic form $Q$ defined over $\mathbb{Z}/n\mathbb{Z}$. Again, this is a wellstudied problem, even for nondiagonal quadratic forms. (See e.g. LidlNiederreiter's "Finite Fields", Chapter 6.) By the Chinese remainder theorem, it is enough to consider $n$ a prime power (and I'll focus, as I said, only on odd prime powers), in the sense that if your function $N(i)$ will be denoted $N(i,n)$ to emphasize the dependence on $n$, then $N(i,n) = \prod_{p^k \mid \mid n} N(i,p^k)$. For odd primes, we have an easy method to lift points modulo $p^{k1}$ to points modulo $p^k$, namely Hensel's lifting, so that $N(i,p^k) = N(i,p^{k1})p^{s1}$ (a modification of this relation is available also for $p=2$). The problem of computing $N(i,p)$ is classical. In this prime case, we in fact have close to squareroot cancellation: $N(i,p)  p^{s1} \le p^{\lceil (s1)/2 \rceil}$ (precise formulas are available, but this inequality is essentially optimal). See Theorems 6.26 and 6.27 in the aforementioned book. This already solves your problem and proves my claim (it even has a slightly better exponent for $p$, compared to the claim, if $s$ is odd)! An application of the CRT shows $$\prod_{p^k \mid \mid n} \left(1 p^{\lfloor \frac{s1}{2}\rfloor} \right) \le \frac{N(i)}{n^{s1}} \le \prod_{p^k \mid \mid n} \left( 1 + p^{\lfloor \frac{s1}{2}\rfloor}\right)$$ and this is essentially optimal. If $s \ge 5$ then the upper bound is bounded by the absolute constant $\prod_{p}( 1 + 1/p^2)<\infty$ and the lower bounded is bounded from below by $\prod_{p}(11/p^2)>0$. If $s \le 4$ this still gives information but, as pointed out by GH, your required estimate is simple false for small $s$. Specifically, it is violated if you take $n$ to be a primorial $\prod_{p \le m}p$ and let $m \to \infty$. It does hold for certain choices of $n$, such as primes, as long as $s \ge 2$. More generally, you can restrict to $n$ with $\sum_{p \mid n}1/p = O(1)$ and then your claim will be true even for $2 \le s \le 4$.
I'll explain how to obtain the claimed estimate without using LidlNiederreiter. A standard way to compute $N(i)$ exactly is to use Fourier analysis on $\mathbb{Z}/n\mathbb{Z}$. (This is related to the random walk interpretation of Emil Jeřábek, and the group structure is useful because it gives simple expressions for the eigenvalues associated with the walk.) Let $e(x):=e^{2\pi i x}$. Then $$N(i) = \frac{1}{n} \sum_{a \bmod n} e\left(\frac{ai}{n}\right) G(a,n)^s$$ where $G$ is the (generalized) quadratic Gauss sum $$G(a,n):=\sum_{x \bmod n} e\left( \frac{ax^2}{n}\right).$$ The main term comes from $a=0$ and is equal to $n^{s1}$. It remains to bound the other terms. The absolute value of $G(a,n)$ is well understood. If $n$ is odd, we have $$\leftG(a,n)\right = \sqrt{n \gcd(a,n)}.$$ Hence, by the triangle inequality, $$\left N(i)  n^{s1} \right \le \frac{n^{\frac{s}{2}}}{n}\sum_{0 \not\equiv a \bmod n} \gcd(a,n)^s = \frac{n^{\frac{s}{2}}}{n} \sum_{\substack{d \mid n\\ d \neq n}} d^{\frac{s}{2}} \phi(n/d)$$ where $d$ stands for $\gcd(a,n)$ and $\phi(n/d)$ counts $a \bmod n$ that have $\gcd(a,n)=d$. If we sum over $e:=n/d$ instead of over $d$, we find that $$\left N(i)  n^{s1} \right \le n^{s1} f_{s}(n)$$ where $$f_{s}(n):= \sum_{\substack{e \mid n \\ e \neq 1}} \phi(e)e^{\frac{s}{2}}.$$ (This inequality is an equality if $i=0$ and $s$ is divisible by $4$, because the only possible loss is when we applied the triangle inequality, but $G(a,n)/\sqrt{n \gcd(a,n)}$ is a power of $\sqrt{1}$.) Once $s\ge 5$, we have the upper bound $$f_s(n) \le \sum_{e \ge p} e^{1\frac{s}{2}} \ll p^{1\frac{s}{2}}$$ where $p$ is the smallest prime divisor of $n$. This establishes the first part of the claim. To see that $N(i) \gg n^{s1}$ for $s \ge 5$, observe that $f_{s}(n)$ is monotonic decreasing and $f_5(n) \le \zeta(3/2)/\zeta(5/2)1\approx 0.94 < 1$.