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 $\tau=1$.
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...
Let $\mu$ be an $n$-AD-regular measure in $\mathbb{R}^d$. Chousionis, Garnett, Le and Tolsa [CGLT] proved that $\mu$ is uniformly $n$-rectifiable if and only if the square function built from the density differences $\Delta_\mu(x,r)=\mu(B(x,r))/r^n-\mu(B(x,2r))/(2r)^n$ satisfies a Carleson condition. In this paper we s...
Let $r_n$ be the infimum of \[ \frac{\lVert P'-P'(0)P\rVert_{H^2}}{\lVert P\rVert_{H^2}} \] over all degree-$n$ polynomials $P$ satisfying $P(0)=1$ whose zeros lie in the closed unit disk. We prove the quantitative residual bound \[ r_n\geq \exp\!\bigl(-(1+o(1))\sqrt n\log n\bigr) \qquad(n\to\infty). \] As an applicati...
Xiaojun Tan, Qihang Wang, Wei Huang et al.· 0 citations
Let $\gamma\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2\pi$ with its curvature $\kappa$, parametrized by arc length, and let $\lambda_\gamma$ be the first eigenvalue of the periodic curvature Schr\"odinger operator $-d^2/d s^2+\kappa(s)^2$. We obtain \[ \lambda_\gamma\geq \frac{\sqrt{\pi}}{2} \left(\f...
Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $\mu$ for Riemannian volume, and set $V(o,r)=\mu(B(o,r))$ for geodesic balls centered at $o$. For $0<\alpha<2$, let $X^{(\alpha)}$ be the process obtained by subordinating Brownian motion with an independent $\alpha/2$-stable subor...
Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\ge 4$, and $\ell = \lfloor\frac{n}{2}\rfloor$. We prove the sharp pointwise estimate \[ q_2(E) \ge -\frac{2(\ell -1)}{3\ell} \mathrm{Scal}(E) \mathrm{Id}_{\Lambda^2V^*} \] for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature...
Jian-Quan Ge· 0 citations
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