For a uniform random Boolean function on p bits, its low-degree Fourier coefficients uniquely determine it with high probability precisely when d exceeds p/2 by an O(sqrt(p log p)) window.
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Low-Degree Fourier Threshold for Random Boolean Functions
For a uniform random Boolean function on p bits, its low-degree Fourier coefficients uniquely determine it with high probability precisely when d exceeds p/2 by an O(sqrt(p log p)) window.