Let $n\geq 2$, and let $\sigma$ denote the standard surface measure on the $n$-sphere $S^{n-1}$, normalized to have total measure $1$. According to the exercise at the end of Terence Tao's blog post Computing Convolutions of Measures, the density $f$ of $\sigma\ast\sigma$ should satisfy $|f(x)|\lesssim (2-|x|)^{(n-3)/2}$ for $1\leq|x|<2$.
I am trying to show this by considering "$\epsilon$-thickenings" of $S^{n-1}$ $A_{\epsilon}:=\{x\in\mathbb{R}^{n} : ||x|-1|\leq \epsilon\}$ and then estimating
$$\frac{|A_{\epsilon}\cap (A_{\epsilon}-x)|}{|A_{\epsilon}|^{2}},$$ where $|\cdot|$ denotes $n$-dimensional Lebesgue measure. According to Tao's claim, I should get the numerator is $\lesssim (2-|x|)^{(n-3)/2}\epsilon^{2}$; however, I am having troubling obtaining the first factor. Any suggestions? I am not interested in exact formulae, orders of magnitude will suffice.