Uma Girish, Avishay Tal, Kewen Wu

We prove that for every parity decision tree of depth $d$ on $n$ variables, the sum of absolute values of Fourier coefficients at level $\ell$ is at most $d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$.

Our result is nearly tight for small values of $\ell$ and extends a previous Fourier bound ...
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