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  "body": "\n## Summary\n\nThe Gowers norms $\\norm{f}_{U^k}$ are norms on complex-valued functions on a finite abelian group, one for each $k \\ge 2$.\nThey average products of $f$ over $k$-dimensional parallelepipeds $x + \\omega_1 h_1 + \\dots + \\omega_k h_k$.\nA function with small $U^k$ norm behaves randomly for counting arithmetic progressions of length $k + 1$.\nGowers introduced them for his proof of [[Szemerédi's theorem|B23B0QEMN5]], and they are the basic objects of higher-order Fourier analysis.\n\n## Explanations\n\n### L1 human\n\nThe Gowers norm tests whether a function, such as the indicator of a set, looks random from the point of view of additive patterns.\nThe $U^2$ norm detects correlation with ordinary waves $e^{2\\pi i a x / N}$, as in usual Fourier analysis.\nThe $U^3$ norm also detects quadratic waves $e^{2\\pi i a x^2 / N}$, which the $U^2$ norm misses.\nIf a set has small $U^{k-1}$ norm after subtracting its density, it contains about as many progressions of length $k$ as a random set of the same density.\n",
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