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.
We prove that there exist two Bloch functions $f_1$ and $f_2$ on $\mathbb D$ such that $$ |f_1(z)|+|f_2(z)| \geq \left(\log\frac{1}{1-|z|}\right)^{1/2}, \qquad z\in\mathbb D, $$ thereby resolving an open problem posed in 2008 by Girela, Pel\'aez, P\'erez-Gonz\'alez and R\"atty\"a. Our proof is based on a new Szeg\H{o}-type recursion involving $\mathbb C^2$-valued polynomials and their reciprocal polynomials.
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
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 application, we answer Erd\H{o}s Problem 973 on exterior power sums in the negative, in a form quantitatively stronger than the answer first obtained by Luo, Yang, and Zhu.
Xiaojun Tan, Qihang Wang, Wei Huang et al.· 0 citations
Let $H_k$ be an $L^2$-normalized Hecke basis for the space of all holomorphic cusp forms of weight $k$. We show that $\max_{f\in H_k}\Vert F\Vert_6\gg (\log\log k)^{\frac{1}{2}}$ where $F(z)=(\Im z)^{\frac{k}{2}}f(z).$ This confirms that the $L^6$-norm of Hecke eigenforms does not converge uniformly as the weight goes to infinity. We also give some results on the joint mass of degree $6$.
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 $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for the $L^1$-mean of the M\"obius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an $\ell^2$-estimate involving the M\"obius function.
B. Doyle· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
