Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors

TL;DR

This paper proves concatenation theorems showing that functions with polynomial or Gowers anti-uniform behavior in separate directions also exhibit such behavior jointly, enabling control of local Gowers norms by global norms.

Contribution

It introduces concatenation theorems for anti-Gowers-uniform functions and characteristic factors, linking separate directional behaviors to joint behavior at higher degrees.

Findings

Establishes concatenation theorems for anti-Gowers-uniform functions.

Shows control of local Gowers norms by global Gowers norms.

Lays groundwork for analyzing polynomial progressions in sets like primes.

Abstract

We establish a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform", "uniformly almost periodic", or "nilsequence") behaviour in two different directions separately, then it also exhibits the same behavior (but at higher degree) in both directions jointly. Among other things, this allows one to control averaged local Gowers uniformity norms by global Gowers uniformity norms. In a sequel to this paper, we will apply such control to obtain asymptotics for "polynomial progressions" $n+P_1(r),\dots,n+P_k(r)$ in various sets of integers, such as the prime numbers.