On a conjecture of Gowers and Wolf

Abstract

Gowers and Wolf have conjectured that, given a set of linear forms $\{{\psi }_i{\}}_{i=1}^t$ each mapping ${\mathbb{Z}}^D$ to $\mathbb{Z}$, if $s$ is an integer such that the functions ${\psi }_1^{s+1},\dots ,{\psi }_t^{s+1}$ are linearly independent, then averages of the form ${\mathbb{E}}_{\mathbf{x}}{\prod }_{i=1}^{t}f({\psi }_i(\mathbf{x}))$ may be controlled by the Gowers $U^{s+1}$-norm of $f$. We prove (a stronger version of) this conjecture.

13 pages; minor changes based on referee comments; formatted for Discrete Analysis

References in corpus (1)

• The true complexity of a system of linear equations

Cited by in corpus (1)

• True complexity and iterated Cauchy--Schwarz