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On the Fourier-Walsh spectrum of the Moebius function

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Abstract

We study the Fourier-Walsh spectrum \(\{ \widehat \mu (S);S \subset \{ 1, \ldots ,n\} \} \) of the Moebius function µ restricted to \(\{ 0,1,2, \ldots ,{2^n} - 1\} \simeq {\{ 0,1\} ^n}\) and prove that it is not captured by levels \(\{ \widehat \mu (S)||S| < {n^{\frac{2}{3} - \varepsilon }}\} \). An application to correlation with monotone Boolean functions is given.

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  1. Institute for Advanced Study, Princeton, NJ, 08540, USA

    J. Bourgain

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  1. J. Bourgain
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Correspondence to J. Bourgain.

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Bourgain, J. On the Fourier-Walsh spectrum of the Moebius function. Isr. J. Math. 197, 215–235 (2013). https://doi.org/10.1007/s11856-013-0002-2

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  • DOI: https://doi.org/10.1007/s11856-013-0002-2

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