The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $\gamma$ is the standard Gaussian in $\mathbb R^n$, $K \subset \mathbb R^n$ is an origin-symmetric convex set, and $s, t \in \mathbb R$ then $\gamma\left(e^{\frac{s + t}{2}} K\right) \ge \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$. Herscovici, Livshyts, Rotem and Volberg proved a stability version of this result, showing that if one has equality up to a factor $(1 + \epsilon)$ in the (B)-inequality for $K$ then the inradius of $K$ must be either ``very large''or ``very small,''where the bounds depend on $\epsilon$ and on $n$. We give a new stability estimate which is dimension-free and also yields more precise information about bodies which are near-optimizers of the (B)-inequality. In particular, our results imply that if $\gamma\left(e^{\frac{s + t}{2}} K\right) \le (1 + \epsilon) \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$, then every principal component of the covariance matrix of the probability measure obtained by restricting the Gaussian to $K$ must either be at least $1 - O(\epsilon)$ or at most $O(\epsilon)$, which is sharp. Our method extends immediately to yield stability estimates for generalizations of the (B)-inequality, namely the ``strong''and ``functional''(B)-inequalities, which reduce to spectral questions about $1$-log-concave measures on $\mathbb R^n$.
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.
Let $A$ be Hermitian positive definite and let $f_m$ denote the Lanczos approximation to $f(A)b$. We prove that if $f(z)$ or $f(z) / z$ is Stieltjes, then the $A^\alpha$-norm error of the Lanczos approximation is within a factor $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ of the the best possible Krylov Subspace Method, where $\kappa(A)$ is the condition number of $A$ and $E = \max\{\alpha,1-\alpha\}$. Our result strengthens and generalizes the upper bound of [Schweitzer; SIMAX, 46.3 (2025)]. Moreover, we prove that the constant $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ is optimal.
We study exponentially damped M\"obius approximants in $\mathscr H=L^2([1,\infty),dt/t^{-2})$. With \[ \gamma_n(t)=\left\lfloor\frac tn\right\rfloor -\frac{\lfloor t\rfloor}{n},\qquad f(u)(t)=\sum_{n\ge1}\mu(n)e^{-nu}\gamma_n(t),\] we compute the relevant scalar products, characterize the M\"obius coefficients as the unique coefficients giving pointwise convergence to the constant function, and prove $\langle1 \mid f(u)\rangle\to1$. Vasyunin's formula expresses $F(e^{-u})=\|f(u)\|_2^2$ as an arithmetic cotangent sum. To analyze $F(x)$ as $x\uparrow 1$, we define the canonical third-order truncation $\mathcal F_{[3]}$ of $F$ by deleting the sole remainder $\rho_3$. We prove exact edge and residue-character cancellations, initial-edge asymptotics, finite-scale formulas, and \[ \mathcal F_{[3]}(x)\ll \frac{\log^2\!\bigl(e/(1-x)\bigr)}{1-x}. \] For the terms containing $\rho_3$, we prove initial-edge asymptotics, and a finite-scale criterion. The unresolved boundedness problem is thereby reduced to explicit global bilinear cancellation.
Let $\omega$ be a concave modulus of continuity that is weaker than Lipschitz, meaning $\omega(t)/t$ diverges as $t$ approaches $0$. We construct a diffeomorphism of the circle with irrational rotation number, in the regularity class $C^{1+\omega}$, with a wandering interval. This construction implies that Denjoy's 1932 theorem is sharp in regularity, unless additional restrictions are imposed on the rotation number. The construction in the special case $\omega(t) = t\log(1/t)$ settles an open problem dating back to Herman's 1979 work on circle diffeomorphisms, which gave constructions for $\omega(t) = t\log(1/t)^{1+\varepsilon}$ for every $\varepsilon>0$. Our examples arise as limits of periodic circle diffeomorphisms with rapidly converging rotation numbers.
Let \[ \mu(r)=\frac{\pi}{2}\frac{\Kc(r')}{\Kc(r)}, \qquad r'=\sqrt{1-r^2},\qquad 0<r<1, \] be the Gr\"otzsch modulus function. Problems 3.38(a)--(b) in a survey of Vuorinen ask whether Newton's iteration for $\mu^{-1}(y)$, initialized by $x_0=1/\cosh y$, converges for every $y>\pi/2$, and whether it is strictly increasing when $y>\pi$. We prove both assertions. The main point is that $\mu$ has exactly one inflection point $a\in(1/2,1/\sqrt2)$. If the zero lies in the convex region, the Newton iterates increase to it from the left. If the zero lies in the concave region, the orbit crosses the inflection point, overshoots the zero at most once, and then decreases to the zero. For $y>\pi$ we obtain the stronger estimate $0<x_n<x_{n+1}<\mu^{-1}(y)<3-2\sqrt2<1.$
Let $\alpha$ be irrational and let $q_j$ be the denominators of its continued-fraction convergents. We study the function \[ h(x)=\sum_{j\geq1}\frac{\cos(2\pi q_jx)}{q_j} \] and the skew product \[f(x,y)=(x+\alpha,y+h(x))\quad\mathrm{mod}\quad\mathbb{Z}^2.\] The function $h$ is H\"older continuous of every exponent below one. A Fourier argument shows that $h$ is not Lipschitz. The map $f$ is a toral pseudo-rotation with rotation vector $(\alpha,0)$, but it has neither bounded mean motion nor $C^0$-rigidity. Suppose $\alpha$ satisfies the Diophantine condition $\mathrm{DC}(\tau)$. Then, $f$ has $(C,1-1/\tau)$-deviation when $\tau>1$; and it has $(C_\delta,\delta)$-deviation for every $0<\delta<1$, but not for $\delta=0$ when $\tau=1$.
