Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher signs and let $a=(a_1,\ldots,a_n)\in\R^n$ satisfy the normalization below. For the normalized Rademacher sum, we determine how its higher moments depend on the fourth-order mass. Combining a sharp fixed-q moment envelope with a separate argument below the convexity threshold gives the Gaussian stability inequality for the full range $p\geq4$ of this linear-in-q bound. The same fourth-order framework determines the sharp finite dimensional $L_p/L_4$ Khintchine constant for $p\geq5$, with the flat coefficient vector as the extremizer. These results settle the conjectures of Jakimiuk and of Bara\'nski, Murawski, Nayar, and Oleszkiewicz stated below. We also prove Jakimiuk's conjectured quadratic stability estimate at $p=3$. The resulting bounds retain information about sparsity and effective dimension, with applications to Rademacher random projections and randomly signed errors; those applications are not developed further here. Their Laplace-transform form also gives coefficient-sensitive tail bounds. The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol.
We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, where $\Phi_1$ and $\Phi_2$ are entire functions of exponential...
The Johnson--Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O(\varepsilon^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\varepsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameter...
Let $1\le m<n$ and let $u$ be an $m$-subharmonic function on a domain in $\mathbb{C}^n$. We study local exponential and polynomial integrability, with particular attention to the sharp polynomial exponent predicted by B{\l}ocki's conjecture. Explicit radial examples show that direct analogues of the Guan--Zhou strong o...
Let $P (Y_1, ..., Y_d)$ be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along $\Omega (|P (n_1, ..., n_d)|)$. Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergels...
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every...
Let $S_k^\sigma(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $\sigma$, and $S_k^{\mathrm{new},\sigma}$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^\sigma(N)$ and $S_k^{\mathrm{new},\si...
Timothy Nelson, Erick Ross, Maya Wassercug et al.· 0 citations
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MIT News · Artificial Intelligence· news.mit.eduOct 23, 2026
Assistant Professor Benjamin Lindquist studies how engineers deliberately injected randomness into early AI, giving rise to deep-learning models that sometimes hallucinate.
