Behavior of $I(n) = \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}e^{ i n (x+y)}\,\frac{2\sin^2(x)\sin^2(y)}{2k -\cos(x)-\cos( y)}\,\mathrm{d}x\,\mathrm{d}y$
Solution 1:
I've proven the asymptotics in Mathoverflow 353430, for large integer $n$
$$ J(n,\kappa) := \Big(\frac{2}{\pi}\Big)^2 \int_{-\pi}^\pi \int_{-\pi}^\pi \exp{(i\,n(x+y))}\frac{\sin^2x\,\sin^2y} {2\kappa - (\cos{x}+\cos{y}) }\, dx \,dy \sim$$ $$ \sim \frac{8}{\sqrt{\pi \kappa n}}(\kappa^2-1)^{7/4} (\kappa - \sqrt{\kappa^2-1})^{2n}\quad, \quad (\kappa>1)$$