Abstract:Recently, Chen \cite{Chen2026} proved that Talagrand's Boolean convolution conjecture holds up to the dimension-free factor \((\log\log\eta)^{3/2}\), namely for every fixed \(\tau>0\), \[
\mu\{P_\tau f>\eta\|f\|_1\}
\le C_\tau
\frac{(\log\log\eta)^{3/2}}{\eta\sqrt{\log\eta}},
\qquad \eta>e^3. \] We revisit the terminal testing-discrepancy step in Chen's perturbed reverse-heat coupling. Chen estimates this discrepancy globally in terms of the remaining gap to the terminal level. We keep the same coupling and the same reverse-heat formulations, but localize the terminal discrepancy on each remaining-gap layer before summing the layers. This changes the fixed-time anti-concentration cost from order \((\log L)^{3/2}/\sqrt L\) to order \((\log L)/\sqrt L\), where \(L=\log\eta\). Consequently, we obtain a \((\log\log\eta)^{1/2}\) improvement as \[
\mu\{P_\tau f>\eta\|f\|_1\}
\le C_\tau
\frac{\log\log\eta}{\eta\sqrt{\log\eta}},
\qquad \eta>e^3. \]
From: Yanjin Xiang [view email]
[v1]
Wed, 3 Jun 2026 08:09:22 UTC (19 KB)
[v2]
Sat, 13 Jun 2026 13:09:27 UTC (18 KB)
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