= Generalized von Neumann inequality for four-term progressions
{title2=$|\mathbb E_{x,d}\prod_{j=0}^3f_j(x+jd)|\leq\min_j\|f_j\|_{U^3}$}
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.
Back to article page
