Orthogonality of integer Fourier modes (${\int }_0^1{e}^{2\pi imx}dx=1_{m=0}$)

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.