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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.

A polynomial phase is a complex-valued function of the form $f(x)=e^{2\pi iP(x)}$, where $P$ is a real polynomial, or a suitably defined polynomial taking values in $\mathbb{R}/\mathbb{Z}$. Its absolute value is one. On a finite cyclic group, a typical well-defined example is $f(x)=e^{2\pi ia{x}^2/N}$ with integer $a$ and $x\in \mathbb{Z}/N\mathbb{Z}$.

The multiplicative derivative satisfies $f(x+h)\overline{f(x)}=e^{2\pi i(P(x+h)-P(x))}$. An ordinary additive difference lowers the degree of a nonconstant polynomial by at least one. Consequently, if the degree is less than $k$, every $k$-fold multiplicative difference is one, and the normalized Gowers uniformity norm $U^k$ equals one on a finite abelian group. This explains why polynomial phases serve as structured comparison functions in higher-order Fourier analysis.