Periodic nilsequences and inverse theorems on cyclic groups

The inverse theorem for the Gowers norms, in the form proved by Green, Tao and Ziegler, applies to functions on an interval $[M]$. A recent paper of Candela and Sisask requires a stronger conclusion when applied to $N$-periodic functions; specifically, that the corresponding nilsequence should also be $N$-periodic in a strong sense. In most cases, this result is implied by work of Szegedy (and Camarena and Szegedy) on the inverse theorem. This deduction is given in Candela and Sisask's paper. Here, we give an alternative proof, which uses only the Green--Tao--Ziegler inverse theorem as a black box. The result is also marginally stronger, removing a technical condition from the statement. The proof centers around a general construction in the category of nilsequences and nilmanifolds, which is possibly of some independent interest.