= 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.<Higher order Fourier analysis by Terence Tao>[Preface and sections 1.3–1.5]{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>.
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