We prove an infinitesimal circular version of Morera's theorem. Let $D\subset\mathbb{C}$ be a domain and let $f\in C(D)$. If, at every $a\in D$, $\int_{\vert{}\zeta-a\vert{}=r}f(\zeta)\,d\zeta=o(r^2)$ as $r\to0^+$, then $f$ is holomorphic in $D$. In particular, exact vanishing of all sufficiently small centered circular integrals implies holomorphicity. The proof uses a local distributional $\partial$-primitive, a circular identity for weak $\partial$-derivatives, and a pointwise asymptotic mean-value criterion for harmonicity.
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.
We prove that for every fixed $\lambda>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\geq1$ satisfy $\max_{2\leq k\leq n+1}|\sum_j z_j^k|>e^{-\lambda n}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erd\H{o}s 973 can exist. The proof combines a truncated exponential factorization with overconvergence on an open set outside the unit disk and a normal-family obstruction for Cauchy transforms.
Over any infinite field we prove existence of polynomial automorphism with prescribed differentials at points. More precisely, let $\K$ be an infinite field and $a_1,\ldots, a_k;$ $b_1,\ldots , b_k$ be two sequences of different points in~$\K^n$, $n\ge 2$. For any sequence $L_1,\ldots, L_k\in SL(n,\K)$ there exists a polynomial automorphism $\Phi: \K^n\to \K^n$ with jacobian one such that $\Phi(a_i)=b_i$ and $d_{a_i} \Phi= L_i$ for $i=1,\ldots, k.$
Zbigniew Jelonek, Gustavo Menani, Maria Michalska· 0 citations
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 Bergelson and Richter: if $P$ is an irreducible binary cubic form and $ (X, T)$ is a uniquely ergodic system with unique invariant measure $\mu$, then for any $x \in X$ and $f \in C(X)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac 1 {N^2} {\mathop{\sum\sum}_{n_1, n_2 \leqslant N}} f \big( T^{ \Omega (|P (n_1, n_2)| ) } x \big) = \int_{X} f \ \mathrm{d} \mu . \end{equation*} Moreover, we prove in the appendix a related conjecture of C\'espedes and Donoso over number fields.
We construct, for any $p\in(1,\infty)$, $p$-homogeneous rank-one convex integrands $F\colon\mathbb R^{2\times m}\to \mathbb R$ that are nowhere quasiconvex when $m$ is large. When $p$ is sufficiently close to $4$, such examples can be constructed on $\mathbb R^{2\times 4}$. Related constructions give, for every $p$, conjugation- and transposition-invariant examples on $\mathbb R^{d\times d}$, and examples on $\mathbb R^{4\times 2}$ for every $p\neq 2$.
We prove an analogue of Gr\"unbaum's inequality on the sphere. Let $n \geq 3$ and let $K$ be a convex body on $\mathbb S^{n-1}\subset \mathbb R^n$ with centroid at $\theta\in \mathbb S^{n-1}$. Then for any $u\in \mathbb S^{n-1}$ that is orthogonal to $\theta$ we have $$\sigma(K\cap u^+) \ge \left(1-\frac{1}{n}\right)^{n-1} \sigma(K),$$ where $\sigma$ denotes the spherical measure. The constant in this inequality is optimal.
S. Myroshnychenko, D. Ryabogin, K. Tatarko et al.· 0 citations
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