Aug 2026· 2 citations· ⚡ 1 influential· 20 references
Mathematics
Abstract
Let $C_q$ denote the group of the $q$th roots of unity. A question arising from the work of Becker, Klein, Slote, Volberg and Zhang is whether the dimension-free Bohnenblust--Hille constants for functions on $C_q^N$ grow subexponentially with the degree. We answer this question affirmatively. In fact, we prove a stronger estimate for functions whose Fourier characters involve at most $d$ coordinates. If $\BHint{d}{q}$ is the optimal constant for this larger class, then, for every fixed $q\geq2$, \[ \BHint{d}{q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right), \] where $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\geq3$. As an application, we obtain two-sided estimates for the Bohr radius of the Fourier layer formed by characters involving exactly $d$ coordinates, and we determine its asymptotic behaviour in natural joint regimes of $d$ and $N$.
Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust--Hille constants govern...
For an $m$-homogeneous polynomial on $\mathbb C^n$, let $D_{m,n}$ denote the optimal constant in the complex polynomial Bohnenblust--Hille inequality, and set $D_m:=\sup_{n\ge1}D_{m,n}$. We prove that the dimension-free constants $(D_m)$ have at most polynomial growth: there are absolute constants $B_0,K<\infty$ such t...
D. Pellegrino, Eduardo V. Teixeira· 3 citations· ⚡3
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 $X_{G_h}(\mathbf{x};q)=\sum_{\mu\vdash n}c_\mu(q)e_\mu(\mathbf{x})$ be the chromatic quasisymmetric function of the natural unit interval graph attached to a Hessenberg function $h$. We establish an infinite class, valid in all orders, for which every nonzero polynomial $c_\mu(q)$ has a nonnegative, log-concave coe...
Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $\lambda>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geq\lambda \ri...
Thang Pham, A. Pinamonti, Dung The Tran et al.· 0 citations
Let $D$ be an odd fundamental discriminant, with $D=1$ permitted, and let $r\geq 1$ be fixed. We prove that, for every sufficiently large integer $\ell$ satisfying $(-1)^\ell D>0$, the first $r$ traced diagonal Rankin--Cohen brackets \[ \mathrm{Tr}_1^{|D|}[G_{\ell-2e,D},G_{\ell-2e,D}]_{2e}, \qquad 1\leq e\leq r, \] are...
Ramin Takloo-Bighash· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
