= Low-degree Fourier mass of a sparse Boolean set
{title2=$\sum_{|S|\leq2}\widehat{1_A}(S)^2\leq9\alpha^{3/2}$}
For a subset of the unbiased <Boolean hypercube> of density $\alpha$, use normalized <Fourier-Walsh transform> coefficients. Applying <Beckner's inequality> with $\rho=1/\sqrt3$ gives $\|T_\rho1_A\|_2^2\leq\|1_A\|_{4/3}^2=\alpha^{3/2}$. On levels at most two the weights $\rho^{2|S|}$ are at least $1/9$, giving the displayed bound. <Parseval's identity> gives total Fourier mass $\alpha$, so when $9\sqrt\alpha<1/2$ the higher levels contain strictly more than the lower levels.
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