Orthogonality of integer Fourier modes
= Orthogonality of integer Fourier modes
{title2=$\int_0^1e^{2\pi imx}\,dx=1_{m=0}$}
For an integer $m$, direct integration gives zero unless $m=0$, in which case it gives one. Products of this identity on the unit cube turn <integrals> of finite <exponential sums> into counts of integer solutions. This is the elementary orthogonality used in the <Vinogradov mean value>.