I have been working on this problem for a while but have no clue at all.
Fix a smooth cutoff function $\varphi: \mathbb R/\mathbb Z\rightarrow [0,1]$ supported on $[−\varepsilon-\delta, \varepsilon+\delta]$ and identically equal to $1$ on $[−\varepsilon, \varepsilon]$. For each prime $p$ define $F(x) = \varphi(\frac{x^2}{p})$ (this roughly locates those $x$ such that $x^2$ has a representative within $\varepsilon p$ of zero).
(a) Show that $\left|\frac{1}{p}\sum_{x\pmod p}F(x)-\varepsilon\right|\leq \delta+O(\frac{1}{\sqrt{p}})$ and that for $k\not\equiv 0 \pmod p$, $\frac{1}{p}\sum_{x\pmod p}F(x)e_p(-kx)= O_\varphi(\frac{1}{\sqrt{p}})$, where $e_p(x)=e^{2\pi ix/p}$.
(b) Let $A_\varepsilon \subset \mathbb Z/p\mathbb Z$ be the set of $x$ such that $x^2$ has a representative within $\varepsilon p$ of zero. Show that $A_\varepsilon$ has density $\varepsilon+O(\delta)$ and has $\varepsilon^3 p^2 +O(\delta)p^2 +O_{\delta,\varepsilon}(p^{3/2})$ $3$-APs (Arithmetic Progressions).
(c) Establish the identity $x^2 − 3(x + d)^2 + 3(x + 2d)^2 − (x + 3d)^2 =0$ and conclude that if $x, x + d, x + 2d \in A_{\varepsilon/7}$ then $x + 3d \in A_{\varepsilon}$ and hence that the number of 3APs in $A_{\varepsilon}$ is $\geq C\varepsilon^3 p^2$.
Any comment is aprreciated.