For a sign-valued table $F$, take the union of supports $S$ with $|S|\leq\ell$ and $\widehat F(S)^2\geq\ell2^{-t}$. This decoding set has size at most $2^t$ by Parseval and the counting bound. Taking $t=\varepsilon s$ gives the set in equation (2).

After averaging outside any set $U$, the decoding set can only shrink, and every surviving point lies in $U$. Consequently the decoding set for the original table serves every side condition; it need not be chosen anew after the condition is known. This is the inclusion used immediately after equation (18) in the proof of Theorem 4.17.