 Gowers U3 norm on an interval

Normalize the conjugated cube average by the number of integer cubes whose vertices all lie in the interval. Equivalently, extend $f$ by zero to a cyclic group of order greater than $8N$ and divide its Gowers U3 norm by that of the interval's indicator function. A quadratic phase then has norm one. Omitting the denominator gives a different but uniformly equivalent interval normalization.