Higher-order Fourier analysis

Higher-order Fourier analysis studies functions using nonlinear phase patterns and Gowers uniformity norms, extending the linear characters used in ordinary Fourier analysis. Its motivation includes counting long arithmetic progressions: controlling ordinary Fourier coefficients does not control every higher-order additive pattern.^[ref]

For example, let $p$ be an odd prime number and put $f(x)=\exp (2\pi i{x}^2/p)$ on the finite field ${\mathbb{F}}_p$. Its multiplicative difference $f(x+h)\overline{f(x)}=\exp (2\pi i(2hx+h^2)/p)$ has a linear phase; three successive multiplicative differences are identically one. Consequently its $U^3$ norm is one, although each normalized ordinary Fourier coefficient has magnitude $p^{-1/2}$. The quadratic phase is highly structured even when the linear probes are small. Higher-order inverse theorems describe the structured objects detected by large Gowers uniformity norms.