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For every $\delta > 0$, $f:\{0,1\}^n\to\{0,1\}$, and $\epsilon >0$, if $\mathsf{H}^{1-\delta}_{\text{avg}}(f)\geq S$, then there is a density-$\delta$ distribution $H$ s.t. for every circuit $C$ of size at most $\frac{\epsilon^2 S}{100n}$, $$ \text{Pr}_{x\in_R H} [C(x)=f(x)]\leq \frac{1}{2}+\epsilon. $$ Exposition : The Impagliazzo Hard-Core-Set Theorem (Luca Trevisan):...