Torquato and Stillinger conjectured an exponential improvement of Minkowski's classical lower bound on the maximal density of sphere packings in high-dimensional Euclidean space $\mathbb{R}^d$ using a pair-correlation-function optimization framework. Conditional on their realizability conjecture, we show that a simple family of hyperuniform pair correlation functions yields polynomial improvements over Minkowski's lower bound of the form $\phi_{\mathrm{max}} \gtrsim d^\beta 2^{-d}$ for every fixed $\beta>1$ in sufficiently high dimensions. As the polynomial exponent is allowed to increase with dimension, this family continuously approaches the previously conjectured exponential improvement. We further derive the same exponential asymptotic rate independently from the Cohn--Elkies dual linear programming upper bound formulation, demonstrating that its radial objective test functions cannot asymptotically exclude packings with the Torquato--Stillinger density scalings. The agreement between these alternative approaches provides new evidence that exceptionally dense disordered sphere packings may exist in high dimensions and strengthens the case for the Torquato--Stillinger conjectural lower bound.
For the purposes of proving Smale's mean value conjecture, one may restrict consideration to an explicitly described family of so-called Schlicht normalized polynomials. For Schlicht normalized polynomials whose leading coefficient decays sufficiently slowly, we show that the conjectured upper bound $1$ becomes asymptotically valid as the degree $d$ tends to infinity. The key and novel input is a refined Koebe $\frac{1}{4}$-type theorem of Cunningham, tailored to Schlicht functions whose images have bounded logarithmic capacity. For the dual mean value conjecture, we obtain an improvement of Eremenko's Markov-type inequality for regions bounded by polynomial lemniscates on which the polynomial is univalent. As a consequence, we improve the best known unconditional lower bound $\frac{1}{d^2}$ of Dubinin to $\frac{(d-\frac{1}{2})^{1/d}}{d^2}$ for all $d \ge 2$. The strengthened Markov-type inequality constitutes one of the main technical contributions of this paper. Its proof combines the solution to a related extremal problem for the logarithmic capacity of polynomial lemniscates, with the quasi-conformal deformation method introduced by Eremenko and Hayman to establish the connectedness of extremal polynomial lemniscates arising in the Erdos, Herzog and Piranian's problem on maximal lemniscate length.
Wirsching (2003) reduces uniform positive predecessor density for the $3n+1$ map to a chain of five conditions, organized into three conjectures. We prove two of them. Conjecture 1 concerns his path-counting generators: in the convolution carrying them to his Elka functions the partition weights grow subexponentially while the binomial ratio decays geometrically, so a window of width $O(\sqrt{\ell})$ dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density $\varphi$, a base-3 analogue of the Fabius density, against an explicit $\varphi_0$ due to Berg and Kr\"uppel. The exact log-Laplace transform of $\varphi$ splits into a smooth part, a log 3-periodic correction $H$, and a doubly exponentially small remainder. Berg and Kr\"uppel represented that correction as an infinite product in 1998; their analysis did not determine whether it is constant. We give $H$ as a Fourier series with coefficients in closed form in $\Gamma$ and $\zeta$; a classical zero-free theorem for $\zeta$ shows it is not constant, and we enclose its oscillation rigorously. Wirsching's comparison class fixes one phase of $H$, and there Conjecture 3 holds with limit $e^{H(0)}$, certified to lie in $(0.53412203666478,0.53412203666479)$. Off that class the phase sweeps a full period, so the unrestricted asymptotic $\varphi(t)\sim\kappa\varphi_0(t)$ fails. Condition $(\star4)$ follows, at every window radius, with $\mu=1/3$: what Wirsching's argument needs is weaker than Conjecture 3 itself, and the same saddlepoint chain settles it directly. With Conjecture 1 the chain reduces to the single condition $(\star3)$. Conjecture 2 is his route to it and remains open.
Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $\alpha_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $\alpha_0$ is not attained by any finitely supported distribution.
Andrii Arman, A. Bondarenko, A. Prymak et al.· 0 citations
In their work on sphere packing and the conformal bootstrap, Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini conjectured the exact high-dimensional exponent of the Cohn--Elkies sphere-packing linear program. OpenAI's Chapter 1 subsequently proved their conjecture by establishing that both Fourier sign-uncertainty radii are $(1/\pi+o(1))\sqrt d$. We prove the binary coding analogue: the half-rate point of the asymptotic binary Delsarte linear program is $1/2-1/\pi$; equivalently, \[ R_D\!\left(\frac12-\frac1\pi\right)=\frac12. \] We also formulate the two Krawtchouk sign-uncertainty problems and determine both of their asymptotics. If $A^{\mathrm K}_{\pm}(n)$ denotes the first radial layer after which an origin-vanishing Krawtchouk $(\pm1)$-eigenfunction can be nonnegative, then \[ \frac{A^{\mathrm K}_{\pm}(n)}n\longrightarrow \frac12-\frac1\pi. \] The common lower bound is the Hamming space counterpart of the mass-concentration principle in the Chapter 1 proof. The upper bound has a different source. It is the binary-code counterpart of the final spherical-code construction in OpenAI's Chapter 2. Gay, Jeronimo, and Liu improved the resulting binary bound and suggested the functional $\Phi$ used here, but explicitly evaluated only a few low levels of the corresponding hierarchy. We construct and evaluate compatible binary certificates at every level, attaining the upper bound in the limit. The construction uses an $N$-qubit generalization of the pure-state channel of Alrabiah and Guruswami.
Let $S^n$ be the unit round sphere with its intrinsic angular metric, normalized so that $\operatorname{diam}S^n=\pi$. For finite homogeneous metric spaces $X$, put \[ \delta_n=\inf_X d_{GH}(X,S^n). \] The main open problem is whether $\inf_{n\ge2}\delta_n>0$. Gelander's theorem gives $\delta_n>0$ in each fixed dimension, but not uniformly. An abstract cross-polytope construction gives the universal upper bound $\delta_n\le\pi/4$. In the opposite direction, ChatGPT combines the passage from small Gromov--Hausdorff error to an approximate finite action on the sphere, logarithmic stability of approximate inner-product-preserving maps due to Cuesta, operator-norm stability of almost representations, and Green's width theorem for finite transitive sets. This gives the quantitative bound \[ \delta_n\ge \frac{c}{(1+\log(n+1))^2} \] for all sufficiently large $n$.
We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range $p\geq 2$, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.
Unknown authors· 0 citations
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