Generalized von Neumann inequality for four-term progressions ($|{\mathbb{E}}_{x,d}{\prod }_{j=0}^3{f}_j(x+jd)|\le {\min }_j\Vert f_j{\Vert }_{U^3}$)

For functions bounded by one on a cyclic group of prime order greater than three, three applications of Cauchy-Schwarz bound the four-term arithmetic progression average by the smallest Gowers U3 norm of its factors. Zero extension yields the corresponding interval bound with an absolute factor. This makes Gowers U3 norm a measure of uniformity relevant to four-term arithmetic progression counts.