相关论文: p-adic Welch Bounds and p-adic Zauner Conjecture

Let $HD_d(p,q)$ denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in $\mathbb{R}^d$ which satisfy the $(p,q)$-property ($p \geq q \geq d+1$). In a celebrated proof of the…

Let T : Lp --> Lp be a contraction, with p strictly between 1 and infinity, and assume that T is analytic, that is, there exists a constant K such that n\norm{T^n-T^{n-1}} < K for any positive integer n. Under the assumption that T is…

We obtain some sharp estimates for the $p$-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the…

We investigate a complex analogue of Spencer's Six Standard Deviations Theorem. Specifically, we propose the following conjecture: for any dimension $n \geq 2$, given vectors $a_1, \ldots, a_n \in \mathbb{C}^n$ satisfying $\|a_i\|_{\infty}…

Inspired by the recent work of Chen-Sti\'enon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincar\'e-Birkhoff-Witt type results to hold. The tools…

The p-adic description of Higgs mechanism in TGD framework provides excellent predictions for elementary particle and hadrons masses ([email protected] 9410058-62). The gauge group of TGD is just the gauge group of the standard model so…

We show the optimal coherence of $2d$ lines in $\mathbb{C}^{d}$ is given by the Welch bound whenever a skew Hadamard of order $d+1$ exists. Our proof uses a variant of Hadamard doubling that converts any equiangular tight frame of size…

Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type…

The $p$-adic Littlewood conjecture (PLC) states that $\liminf_{q\to\infty} q\cdot |q|_p \cdot ||qx|| = 0$ for every prime $p$ and every real $x$. Let $w_{CF}(x)$ be an infinite word composed of the continued fraction expansion of $x$ and…

We suggest a tight inequality for norms in $d$-dimensional space $l_p $ which has simple formulation but appears hard to prove. We give a proof for $d=3$ and provide a detailed numerical check for $d\leq 200$ confirming the conjecture. We…

We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~\, for the frame potentials $\sum_{ij} \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell $ on a unit--sphere in $d$ dimensions. Sidelnikov's…

Recent findings by Jahn, T. Ullrich, Voigtlaender [10] relate non-linear sampling numbers for the square norm to quantities involving trigonometric best $m-$term approximation errors in the uniform norm. Here we establish new results for…

For any $m \geq 1$, let $H_m$ denote the quantity $H_m := \liminf_{n \to \infty} (p_{n+m}-p_n)$, where $p_n$ denotes the $n^{\operatorname{th}}$ prime; thus for instance the twin prime conjecture is equivalent to the assertion that $H_1$ is…

We verify that $\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx||<\epsilon$ for all real $x$, small primes $p$ and relatively small $\epsilon$. This result supports the famous $p$-adic Littlewood conjecture which states that the above lower…

We generalize to webs of any codimension results already known in codimension one. Given a holomorphic $d$-web $\cal W$ of codimension $q$ $(q\leq n-1)$ in an ambiant $n$-dimensional holomorphic manifold $U$, we define for any integer $p$…

In this article we obtain an "off-diagonal" version of the Fefferman-Stein vector-valued maximal inequality on weighted Lebesgue spaces with variable exponents. As an application of this result and the atomic decomposition developed in [12]…

Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is unbounded. We conjecture that every natural number…

We derive bounds and asymptotics for the maximum Riesz polarization quantity $$M_n^p(A) := \max_{{\bold x}_1, {\bold x}_2, \ldots, {\bold x}_n \in A} {\min_{{\bold x} \in A}{\sum_{j=1}^n{\frac{1}{|{\bold x} - {\bold x}_j|^{p}}}}}$$ (which…

This paper considers overdetermined boundary problems. Firstly, we give a proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two dimensional case and under an additional condition for the case of…

Let $A$ be a central division algebra of prime degree $p$ over $\mathbb{Q}$. We obtain subconvex hybrid bounds, uniform in both the eigenvalue and the discriminant, for the sup-norm of Hecke-Maass forms on the compact quotients of…