We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $\zeta_n:=\arccos(-\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\mathbb{S}^n$. We prove that $$ d_{\mathrm{GH}}(\mathbb{S}^n,\mathbb{S}^{n+1})=\frac{\zeta_n}{2} \qquad(n\geq1), $$ resolving a conjecture of Lim, M\'emoli, and Smith. All cases $n\geq4$ were previously open. This equality is established by explicitly constructing a family of correspondences $\mathcal R_n\subseteq \mathbb{S}^{n+1}\times \mathbb{S}^n$, whose distortion matches the known quantitative Borsuk-Ulam lower bound $\zeta_n$. We also introduce synchronized spherical joins and suspensions of correspondences and prove that the distortion of a join is exactly the maximum of the distortions of its factors. In particular, suspension preserves distortion. Applying these join and suspension operations to the optimal correspondences $\mathcal R_n$ yields new bounds for spheres of nonconsecutive dimensions, including $$ \lim_{m\to\infty} d_{\mathrm{GH}}\bigl(\mathbb{S}^m,\mathbb{S}^{m+d(m)}\bigr) = \frac{\pi}{4} \qquad\text{whenever } d(m)\geq1,\ \text{and }d(m)=o(m).$$
What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by $d_\gh(B^m,B^n)$, for $m>n$? Note that the lower bound coming from the stability of persistent homology is zero, since all balls possess identical (trivial) persistent homology. To establish non-trivial lower bounds, we exploit the Borsuk--Ulam theorem. For any $n \ge 1$, we prove that $d_\gh(B^m,B^n)\ge \frac{\sqrt{n+1}}{\sqrt{n+1}+\sqrt{n}}>\frac{1}{2}$ for $m>n$, and that $d_\gh(B^m,B^n)\to 1$ as $m\to \infty$. Finally, we prove that $d_\gh(B^m,B^n)<1$ for all finite $m>n\geq 1$.
For a finite set $A \subset \mathbb{R}_{>0}$ and a finite graph $H$, let $\chi_H(\mathbb{R}^n;A)$ be the minimum number of colors required to color $\mathbb{R}^n$ while avoiding a monochromatic copy of $H$ whose edges have distances in $A$. Extending the graph-copy framework of Axenovich, Liu, and Sagdeev and a multiple distance theorem of Naslund, we prove for any positive integer $m$, \[\chi_H(\mathbb{R}^n;m):=\max_{\substack{A \subseteq \mathbb{R}_{>0} \\ |A|=m}} \chi_H(\mathbb{R}^n;A) \geq \left(\Gamma_{\chi}\sqrt{\frac{m+1}{\Xi(H)}}+o(1)\right)^n.\] Here, $\Gamma_{\chi}$ is a constant and $\Xi(H)$ is an explicit structural parameter that can be substantially smaller than $|V(H)|-1$, thereby recovering Naslund's similar bound for complete graphs and improving the general bound inherited from the corresponding clique for many graph families. Along the way, we construct a weighted strengthening of the semi-diagonal flattening rank theorem of Correia, Sudakov, and Tomon.
Ayşegül Kula, M. Omar, Jonah Stockwell et al.· 0 citations
In this paper, we establish the boundedness of the bilinear Bochner-Riesz means $\mathcal{B}^{\alpha}_R$ on the complex sphere $\mathbb{S}$. More precisely, we prove that $\mathcal{B}^{\alpha}_R$ is bounded from $L^{p_1}(\mathbb{S}) \times L^{p_2}(\mathbb{S}) \to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\leq p_1, p_2 \leq \infty$, for an admissible range of exponents, with the required smoothness parameter $\alpha$ described in terms of the topological dimension of $\mathbb{S}$. To facilitate our proof, we establish several analytic estimates, including restriction-type estimates, weighted Plancherel estimates with large power of weights and bilinear weighted Plancherel estimates, which are derived from the ground up in our setting and may be considered of independent interest.
A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville numbers, there exist orientation-preserving $C^\infty$ diffeomorphisms $f:\mathbb{R}\to\mathbb{R}$, arbitrarily close to the identity and transcendental over $\mathbb{R}(x)$, such that for every real number field $K\subset\mathbb{R}$, every $n\geq 1$, and every $m\geq 0$, \[ D^m(f^{\circ n})(K)\subseteq K, \qquad D^m(f^{\circ n})(\mathscr{L})\subseteq\mathscr{L}. \] In fact, the non-analyticity locus may be prescribed as any nonempty compact perfect nowhere-dense set disjoint from the real algebraic and Liouville numbers. The proof combines Maillet's theorem with an arithmetic refinement of K\"orner's smooth polynomial sewing method and a rational-germ construction.
We prove that for every integer $n\geq 2$ and $8\pi(n-1)<\rho<8\pi n$, the singular Liouville equation $\Delta u+\e^u=\rho\delta_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item $E_{\mathrm{i} b}$ admits no spherical metrics with a conical singularity of angle $2\pi\vartheta$ as long as $\vartheta$ is a positive odd integer. \item For every integer $n\geq 1$, $E_{\mathrm{i} b}$ admits exactly $n$ spherical metrics with a conical singularity of angle $2\pi\vartheta$ for each $\vartheta\in (2n-1, 2n+1)$. \end{itemize} The basic idea is to prove that the linearized equation has only trivial solutions in the space of axisymmetric functions. The previous method of analysing nodal domains via Bol's isoperimetric inequality only works for $\rho\leq 8\pi$. We develop a unified approach for all $\rho\in (0,+\infty)\setminus 8\pi\mathbb{N}_{\geq 1}$ by exploring the deep connection with the monodromy of the classical Lam\'{e} equation.
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 $n$.
I. Benjamini· 0 citations
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