*Sorry, thinking about it some more, both cases will give something rapidly oscillating. If you split up the integrand into a radial and polar part, it should not be too difficult to compute using some standard oscillating integral techniques

You are interested in the dot product of the Fourier series $A(x) = \sum_{j\ge -k} e^{j(ix)}$ and $B(x) = \sum_{j=0}^n \binom{n}{j} e^{j(ix)}.$ There are nice closed forms for both functions, though of course $A(x)$ is singular and should be treated carefully (for example by fudging it and studying $A(x-i\epsilon)$ to get a smooth Fourier series.) The resulting integral will be dominated by its contribution near $x=0;$ depending on $k<<\sqrt{n}$ or $k>>\sqrt{n}$ it will either be a simple or an oscilating integral.

Thanks! Is there a good criterion for when such a result does hold?