FFT grid on a periodic interval ($x_j=jL/N,0\le j<N$)

Use $N$ equally spaced samples on a period of length $L$, excluding the repeated endpoint. Their spacing is $L/N$, and the discrete Fourier transform angular wavenumbers are $2\pi m/L$. Including both endpoints in an $N$-point array gives spacing $L/(N-1)$ and is inconsistent with that wavenumber assignment. It does not supply an extra independent periodic sample.