Research proof index

A dependent-random-choice estimate. For $p \ge 2$, a nonnegative function $f$, nonempty $A$, and intersecting sets $B_1,B_2$, the support hypothesis produces subsets $A_1 \subseteq B_1$ and $A_2 \subseteq B_2$ whose normalized difference convolution has controlled correlation with $f$. Both relative sizes $|A_i|/|B_i|$ are bounded below by the same explicit quantity, namely one quarter of a normalized $2p$-th power of the weighted $L^p$ norm of $1_A \mathbin{\circleddash} 1_A$.

A dependent-random-choice corollary. Let $A$ be nonempty, let $0<\varepsilon\le 1$ and $\delta>0$, and let $p$ be a nonzero even integer satisfying $\varepsilon^{-1}\log(2/\delta)\le p$. Then there are sets $A_1,A_2$ such that the normalized difference distribution $\mu_{A_1}\mathbin{\circleddash}\mu_{A_2}$ assigns mass at least $1-\delta$ to the source's sifted set $s_{p,\varepsilon}(A)$. Both sets retain explicit density: $\operatorname{dens}(A_i)\ge \tfrac14\operatorname{dens}(A)^{2p}$ for $i=1,2$.