Gowers norm

Summary

The Gowers norms $\norm{f}_{U^k}$ are norms on complex-valued functions on a finite abelian group, one for each $k \ge 2$. They average products of $f$ over $k$-dimensional parallelepipeds $x + \omega_1 h_1 + \dots + \omega_k h_k$. A function with small $U^k$ norm behaves randomly for counting arithmetic progressions of length $k + 1$. Gowers introduced them for his proof of Szemerédi's theorem, and they are the basic objects of higher-order Fourier analysis.