Fourth-moment conjectures for Rademacher sums

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Jakimiuk, Stability of Khintchine inequalities with optimal constants (Bernoulli 32, 2026) evidence for this event

Baranski, Murawski, Nayar, Oleszkiewicz - On the optimal Lp-L4 Khintchine inequality evidence for this event

Fourth-moment conjectures for Rademacher sums evidence for this event

VibeMathed record: Fourth-moment conjectures for Rademacher sums evidence for this event

This site's numerical check of Theorems 1.1 and 1.3 and of Proposition 4.3 evidence for this event

Fourth-moment conjectures for Rademacher sums parent of this event

Four results, and the first is partly a refutation. Jakimiuk conjectured $c_p = \mu_p - 1$ is optimal for every $p \ge 3$; the paper proves that for $p \ge 4$ and gives a counterexample for every $2 < p < 4$, so the conjecture is false as posed and the corrected range is $p \ge 4$. The witness is the two-coordinate vector $S_2 = (\varepsilon_1+\varepsilon_2)/\sqrt2$. The Baranski-Murawski-Nayar-Oleszkiewicz flat-point conjecture is proved outright, in the stronger form that $x \mapsto \|x+S_n\|_p/\|x+S_n\|_4$ is strictly decreasing on $[1,\infty)$ for every real $p \ge 5$; that range is the one they conjectured, so nothing is left over. Jakimiuk's second conjecture, dimension-free quadratic stability at $p = 3$, is proved with an explicit constant, though the optimal constant there is only bracketed and stays open. The paper also records the exact fixed-$q$ moment and Laplace-transform envelopes, from which coefficient-sensitive tail bounds follow. parent of this event