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  "body": "\n## Summary\n\nA Littlewood polynomial is a polynomial all of whose coefficients are $+1$ or $-1$.\nOn the unit circle such a polynomial of degree $n$ has mean square $n + 1$, so its typical size there is $\\sqrt{n+1}$.\nLittlewood asked how close to constant its absolute value on the circle can be.\nThis question about flat polynomials links harmonic analysis, combinatorics and signal design.\nThe [[Rudin–Shapiro polynomials|PSAX8W3ZYV]] are a classical family that is flat from above.\n\n## Explanations\n\n### L1 human\n\nWrite a polynomial using only the coefficients $+1$ and $-1$, for example $1 + z - z^2$.\nOn the unit circle $\\abs{z} = 1$, the sum of squares of the coefficients controls the average of $\\abs{p(z)}^2$.\nThe question is whether the signs can be chosen so that $\\abs{p(z)}$ is nearly the same everywhere on the circle.\nSuch flat sign sequences are useful in signal processing, where they spread energy evenly over all frequencies.\n",
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