Asymptotics of an integral

Consider an integral $$ I(x) = \int\limits_{\mathbb{R}^n} e^{i\xi x } \delta(\xi^2-k^2)\chi( (\xi-k,\gamma)) \, d\xi $$ where $x, k, \gamma \in \mathbb{R}^n$, $|\gamma| = 1$ and $(x,y)\equiv xy = x_1 y_1 + \ldots + x_n y_n$. Here $\delta$ is the Dirac delta, $\chi$ is the Heaviside step function: $$ \chi(t) = 1_{\left\{ t \geqslant 0 \right\}}. $$ How can I obtain asymptotics of such integral $I(x)$ as $|x| \to \infty$?

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2022-07-25 20:46:14
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