Papers

By a metacyclic field $K$ we mean the Galois closure of a pure field $\mathbb{Q}(\sqrt[\ell]{D})$, $D\in\mathbb{Z}$, of odd prime degree $\ell$. Let $\mathcal{M}_\ell$ denote the collection of isomorphism classes of metacyclic fields of degree $\ell(\ell-1)$. Write $N_\ell(X)=\#\{K\in\mathcal{M}_\ell: |\Delta_K|\leq X\}$ for the associated counting function, where $\Delta_K$ denotes the discriminant of $K$. We show $$N_\ell(X)\sim A_\ell X^{\frac{1}{(\ell-1)^2}}(\log X)^{\ell-2}\,,$$ for an explicit constant $A_\ell$. We express $A_\ell$ as a rational number times a product over primes of a degree $\ell$ polynomial in $1/p$.

We introduce a method for studying $K$-theoretic positivity on permutohedral toric varieties through the topology of spaces arising in tropical geometry. The key ingredient is the theory of Grothendieck weights developed by the author. We prove two positivity results using this method. The first result is the positivity of the Euler characteristics of tautological bundles associated with an arbitrary matroid and twisted by a nef line bundle. This gives numerical evidence for a conjectural vanishing theorem. The second result generalizes the external activity complex of Berget–Fink, originally defined for a pair of matroids, to the case of any tuple of matroids with no common loop. We deduce a formula for its graded $K$-polynomial in terms of exterior powers of the dual tautological quotient classes of the matroids. After a change of variables, its coefficients alternate in sign. We also prove the Cohen–Macaulayness of each such complex using the vanishing theorems for combinatorial geometries developed by Eur–Fink–Larson. This proof is new even in the case of a pair of matroids. As an application, we interpret certain Chern numbers of tautological quotient classes as counts of facets, partially answering a question of Berget–Eur–Spink–Tseng.

We prove that solutions arising from smooth, sufficiently small, compactly supported perturbations of non-vacuum constant states in three-dimensional irrotational compressible flow must blow up in finite time, without any symmetry assumptions or other restrictions on the initial data. Moreover, we prove that shock inevitably forms at the boundary of the maximal Cauchy development, and that its formation time agrees, in the small-data asymptotic regime, with the lifespan predicted by the radiation-field analysis.

We consider the Langevin diffusion $dX_t = - \beta \nabla V(X_t) dt + \sqrt{2} dB_t$ for a general nonnegative real-analytic potential $V$ and a large parameter $\beta$. In the large-$\beta$ limit the process is confined to the zero set of $V$, assuming that it starts there. We derive a candidate limiting evolution on the zero set. To do so, the zero set is partitioned into strata according to a measure of local codimension known as the local learning coefficient and its multiplicity. It is then shown that the Dirichlet form associated with $X$ converges in a certain sense to a hierarchy of Dirichlet forms corresponding to a stochastic evolution on the strata. This evolution is strongly biased toward higher-dimensional, or "more singular", strata. This result is motivated by a question from Watanabe's singular learning theory regarding the learning dynamics of overparameterized statistical models and the generalization puzzle in deep learning. The result suggests a mechanism for the observation that stochastic gradient methods tend to be biased toward singular solutions that generalize well.

Ensuring the bijectivity of spline-based parameterizations is fundamental in geometric modeling and isogeometric analysis, as invalid mappings may lead to self-intersections, singular Jacobians, and numerical instability. While T-splines offer enhanced flexibility through local refinement, this flexibility also makes bijectivity verification significantly more challenging. In this work, we propose a rigorous and efficient framework for bijectivity analysis of rational T-spline surfaces based on Bézier extraction. The key idea is to reformulate the T-spline representation into a collection of element-wise rational Bézier patches, on which the Gram determinant of the mapping admits a Bernstein polynomial representation. This enables a coefficient-based analysis of local regularity by exploiting the convex hull and positivity properties of the Bernstein basis. Based on this formulation, we derive a sufficient condition for bijectivity from the nonnegativity of Bernstein coefficients, together with a necessary condition based on the sign consistency of corner coefficients. For cases where these conditions are inconclusive, we introduce a hierarchical subdivision strategy that progressively localizes ambiguous regions and resolves them through refinement. The proposed method provides a certified and adaptive procedure for bijectivity verification that avoids dense numerical sampling and remains computationally efficient. Numerical experiments on complex T-spline geometries demonstrate that the approach accurately detects both valid and near-degenerate configurations, while scaling effectively to large models with thousands of rational Bézier patches. The framework is fully compatible with standard isogeometric analysis workflows.

A quantum state is called absolutely separable (resp. absolutely positive partial transpose (APPT)) if it remains separable (resp. positive partial transpose (PPT)) under any global unitary transformation. It is known that the set of all absolutely separable states is a subset of the set of all APPT states. Moreover, these two sets are identical for qubit-qudit systems. In this note, we give an upper bound for the purity of APPT bipartite states (and therefore for the purity of absolutely separable bipartite states). For qubit-qudit systems, this upper bound becomes the maximum purity of APPT (and absolutely separable) states.

We study $b_d(n)$, the minimum number of points needed to meet the relative interior of every $(d-1)$-simplex spanned by an $n$-point set in general position in $\mathbb{R}^d$. In the plane, this is the parameter from the Blocking Conjecture. We improve the best known general planar lower bound to $ b_2(n)\ge \frac{41}{13}n-O\left(\frac{n}{\log n}\right)$. For $n$ points on the moment curve in even dimension $2r$, we prove that at least $\frac{1}{r!}n^r\log n-O_r(n^r)$ points are needed to pierce the relative interior of all its codimension-one simplices, which exceeds the number of codimension-one faces in a triangulation by a $\log n$ factor. For equally spaced points on the moment curve in odd dimensions, we construct an optimal blocking set whose size equals the maximum number of codimension-one faces in a triangulation.

We analyze a Galerkin spectral method applied to the Vlasov-Poisson (VP) system based on time-independent asymmetrically-weighted (AW) Hermite functions in velocity. The VP system is written as an hyperbolic system using AW Hermite functions in velocity. Unlike the classical Petrov-Galerkin spectral method, which is not stable, the proposed Galerkin spectral method admits a natural stability in the unweighted L2 norm. This stability property enables a rigorous convergence analysis of the method. For sufficiently regular solutions with exponential decay in velocity, we establish error estimates between the exact and numerical solutions and prove convergence of the Galerkin spectral method. In particular, the method achieves spectral convergence in Sobolev spaces, with convergence rates determined by the regularity of the exact solution. Since the Gram matrix arising from the proposed Galerkin method is dense, we derive an equivalent form that retains the same approximation properties while preserving the sparsity structure similar to the classical Petrov-Galerkin method.

The Samuelson condition, a classical area-ratio condition for planar Lagrangian $2$-webs, is equivalent in local web coordinates $(u,v)$ to $\partial_u\partial_v\log|J_F(u,v)|=0$, where $F$ is the inverse web-coordinate map and $J_F$ is its Jacobian. For the tangent-line family $L_t: y=tx+h(t)$, with $h$ smooth on an open interval, let $\Psi(u,v)$, for $u\neq v$, be the intersection map of $L_u$ and $L_v$, write $J_{\Psi}$ for its Jacobian, and set $S_h(u,v) := \partial_u\partial_v\log|J_{\Psi}(u,v)|$ where $J_{\Psi}(u,v)\neq0$. Near each diagonal point $(t_0,t_0)$ with $h''(t_0)\neq0$, the obstruction admits the decomposition $S_h(u,v) = \frac{1}{(v-u)^2} + R_h(u,v)$, where $R_h$ extends smoothly across the diagonal near $(t_0,t_0)$. The universal pole yields $S_h(u,v)>0$ off the diagonal in a sufficiently small neighborhood of $(t_0,t_0)$, so the tangent web does not satisfy the Samuelson condition there. We call $R_h(t,t)$ the Schwarzian residue and show that $R_h(t,t) = \frac{1}{2}\{\sigma,t\} = -\kappa_{\mathrm{aff}}(\sigma) \left(\frac{d\sigma}{dt}\right)^2$. Here $\sigma$ is the equi-affine arclength parameter of the envelope $\gamma(t)=(-h'(t),h(t)-t h'(t))$, and $\{\sigma,t\}$ denotes the Schwarzian derivative of $\sigma$ with respect to $t$. The equi-affine curvature $\kappa_{\mathrm{aff}}$ is defined by $\gamma_{\sigma\sigma\sigma}+\kappa_{\mathrm{aff}}\gamma_\sigma=0$. We also derive the transformation law of the residue under reparametrization of the tangent-line family.

For a graph $G$, let $\alpha(G)$ be the second smallest eigenvalue of the Laplacian matrix of $G$, also known as the algebraic connectivity. Algebraic connectivity plays an important role in characterizing the connectivity of graphs and convergence properties of networks. Kolokolnikov conjectured that among all graphs on $n$ vertices with exactly $2n-4$ edges, $\alpha(G)\leq 2$ and one of the maximizers is the complete bipartite graph whose two parts have sizes two and $n-2$, respectively. In this paper, we completely resolve this conjecture.

In this article, we establish the nonlinear time-asymptotic stability of viscous shock wave with arbitrary large amplitude to the Cauchy problem for the compressible magnetohydrodynamics system. In contrast to prior related work, which relied on various smallness assumptions on both the amplitude and the speed of the viscous shock wave, here we show that any viscous shock profile is time-asymptotically stable under small initial disturbance with zero mass condition without making any assumptions on the amplitude and speed of the viscous shock wave. Moreover, while most of the stability results are obtained for constant viscosity in the momentum equation, the present results hold even for the general bulk viscosity as a smooth function of specific volume. The key part of the proof is introducing a new variable which provides an effective diffusion in the specific volume.

The barren plateau (BP) phenomenon poses a fundamental challenge to the trainability of variational quantum eigensolvers (VQEs) by causing exponentially vanishing gradients as the system size increases. While extensive studies have investigated the geometric origins of BP, its impact on the optimization dynamics and complexity of practical algorithms under finite-shot measurements has received comparatively less attention. In this paper, we develop a theoretical framework that characterizes how the BP affects the optimization dynamics of the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm and quantifies the resulting iteration complexity and measurement budget. We derive non-asymptotic bias and variance bounds for the SPSA gradient estimator, introduce a signal-to-noise ratio (SNR) defined as the ratio of the gradient energy to the mean squared error of the estimator, and establish convergence guarantees for SPSA under finite-shot measurements. Our results show that, under the proposed parameter schedules, the number of iterations sufficient to guarantee a fixed relative first-order stationarity level scales polynomially with the inverse of the intrinsic gradient energy. Consequently, when this energy decays exponentially under BP, the resulting guaranteed iteration complexity and total measurement budget both grow exponentially with the number of qubits.

Recently, Hirose proved an analogue of the linear part of Kawashima's relations for refined symmetric multiple zeta values. In this paper, we establish an analogue of Hirose's relation for finite multiple harmonic $q$-series at roots of unity. As an application, we prove the cyclic sum conjecture proposed by Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso.

We introduce a new amenability property for étale groupoids, termed fiberwise amenability, along with a stronger variant termed ubiquitous fiberwise amenability. (Ubiquitous) fiberwise amenability emerges naturally from a coarse-geometric perspective on étale groupoids and, in the special case of transformation groupoids, it coincides precisely with the amenability of the acting group (rather than topological amenability of the action). It is also tightly linked to the existence of invariant measures on the unit space of the groupoid. The coarse-geometric framework for étale groupoids that we develop systematically in this work allows us to establish several foundational properties of (ubiquitous) fiberwise amenability. As an application, we prove a Følner–paradoxical dichotomy for minimal étale groupoids, which will serve as a key tool in a sequel on almost elementariness of étale groupoids.

The goal of the paper is to provide a complete analysis of a stochastic control problem when the cost to minimize involves the running maximum of the underlying controlled state process. The model is motivated by the electricity cost minimization of a data center facing coincident peak charges and possibly benefiting from participation in a demand-response program. From a mathematical standpoint, the main challenge comes from the inclusion of the running maximum in the state of the system forcing the original problem into a singular stochastic control problem on a closed wedge in the plane with reflection on parts of the boundary. We prove that the value function is locally Lipschitz continuous in space using a gap-monotonicity estimate for coupled reflected loads and a deterministic total-variation bound for the difference of two running maxima. The dynamic programming principle and viscosity formulation are then derived for ordinary progressively measurable controls. The corresponding HJB equation includes a reflecting Neumann condition on the lower boundary for the load direction as well as an oblique boundary condition on the diagonal part of the boundary of the domain due to the effect of the running-maximum process. We prove existence of a solution in the viscosity sense as well as a comparison theorem providing conditional uniqueness in the corresponding viscosity class. Finally we provide numerical results illustrating the trade-off between the two conflicting incentives comprising the expected cost of the data center.

Higher-order networks, represented as hypergraphs, enable direct modeling of multi-body interactions of arbitrary size. Hypergraph representations of real-world systems have been observed to exhibit high simpliciality — the tendency for subsets of hyperedges to also appear as hyperedges — yet the generative mechanisms responsible for this structure are poorly understood. We introduce a generalized preferential attachment hypergraph model in which both hyperedge size $Y_t$ and the number of new nodes per step $X_t$ are drawn from arbitrary distributions, and derive analytically, using a mean-field approximate master equation approach, that the stationary hyperdegree distribution follows a power law whose exponent depends only on the ratio $p = E[X_t]/E[Y_t]$, independent of the shapes of the underlying distributions. Crucially, both $X_t$ and $Y_t$ can be estimated directly from any timestamped hypergraph dataset via a backward-stepping procedure, enabling the model to be fit without parametric assumptions. Applying a nonlinear extension of the model to eight real-world hypergraph datasets, we find that the simplicial fraction increases monotonically with the strength of preferential attachment up to the gelation transition at $\alpha > 1$, establishing preferential attachment as a simpliciality-enforcing mechanism.

Gyárfás and Sárközy [Studia Sci. Math. Hungar., 2023] defined a B-coloring of a graph to be a proper coloring of the edge set in which any $C_4$ is totally multicolored. Let $q_B(G)$ denote the minimum number of colors sufficient for a B-coloring of a graph $G$. In this paper, we prove that any planar graph $G$ with $\Delta=\Delta(G)$ and $\Delta_2=\Delta_2(G)$ has $q_B(G)\leq\Delta+\max\{\Delta_2,38\}$, refining a bound by Kong, Wang, and Zheng [J. Graph Theory, 2026].

The Traveling Salesman Problem is a canonical setting for studying how a population-based method should allocate a fixed search budget between exploration and progressively stronger local intensification. We propose ADPSO-ERLS, a discrete swarm algorithm that treats this allocation as an explicit, tunable design variable. It couples memory-guided swap mutation, heterogeneous initialization, selective candidate-restricted 2-opt during evolution, and an incumbent-only final refinement combining candidate-restricted and optional full 2-opt with double-bridge perturbations. The method is PSO-inspired, using personal and global memories yet dispensing with velocity, inertia, and acceleration coefficients. All six algorithms are implemented in Rust, run on identical hardware, and stopped at a strict, recorded limit of 100,000 candidate-solution assessments, so that programming language, hardware, and evaluation budget are held common across methods; wall-clock time is reported separately because equal assessment counts need not correspond to equal arithmetic work. Over 50 runs on five symmetric TSPLIB instances under the integer EUC\_2D convention, ADPSO-ERLS attains the lowest best and mean cost on every instance, with best-tour Gap of $1.93$–$4.17\%$ and relative error of $3.04$–$5.50\%$. It ranks first under the Friedman test, and all twenty-five multiplicity-controlled Wilcoxon comparisons favor it with large, near-complete distributional separation. A paired ablation with common seeds links initialization, in-run local search, and final refinement to quality gains, while candidate restriction chiefly cuts runtime, by up to a factor of roughly $38$. Further experiments up to $16{,}862$ cities keep best-tour Gaps below $6.7\%$, solving the largest case in under eleven minutes.

Learning dynamics in zero-sum games are typically analyzed under algorithmic symmetry: both agents use the same update rule, or methods from a common algorithmic family. This is at odds with the nature of zero-sum games; competing agents need not coordinate on algorithm selection. This paper studies algorithmic asymmetry in learning dynamics in zero-sum games. In particular, we ask whether fast convergence can be recovered when one agent is fixed to vanilla gradient descent, whose standard regret-based analysis certifies, at best, $O(1/\sqrt{T})$ ergodic convergence. We show that the slow rate is not intrinsic. When one agent uses gradient descent, the opposing agent can use a modified optimistic update, which we call Alternating Optimistic Gradient Descent (AOGD), to make the joint dynamics simulate Alternating Gradient Descent on the even iterates. As a result, the time-average of the asymmetric GD vs.\ AOGD dynamics converges to Nash equilibria at rate $O(1/T)$. Our results show that fast convergence need not require coordinated algorithm selection: one agent can compensate for a slower opponent. More broadly, the paper highlights algorithmic asymmetry as a useful lens for understanding cross-class interactions in multiagent optimization.

We introduce gemstones in a pretriangulated category, which are used to perturb pretriangulated structures. For a triangulated category $\mathcal T$ whose stable category of coherent functors admits a gemstone, we construct pretriangulated structures on $\mathcal T$ which fail the octahedral axiom (TR4). This confirms a long-standing conjecture on triangulated categories.

Given a sequence $\mathfrak{a}:=(a_1,\ldots,a_n)$ of reals, a subsequence $\mathfrak{b}=(a_{i_1},\ldots,a_{i_k})$ is said to be "caged" if the largest and smallest among the members of $\mathfrak{b}$ are $a_{i_1}$ and $a_{i_k}$, though not necessarily in that order. In this paper, we consider the problem of maximal caged sequences in permutations $\pi\in S_n$. We also consider the same problem for a random permutation, both when the permutation is chosen uniformly at random and also when it is picked uniformly at random from among the permutations of rectangular shape, via the RSK correspondence. We also give a $O(n\log n)$ time algorithm to obtain these maximal caged sequences. In addition, we propose a randomized linear time algorithm that gives the length of the maximal caged sequence up to the factor of $(1-o(1))$.

We investigate heteroclinical solutions of a vector-valued Bose-Einstein condensation system involving the $p$-Laplacian. The main difficulty comes from the degeneracy of the $p$-Laplacian and the possible nonsmooth behavior of the potential wells. Under suitable assumptions on the double-well potential, we establish the existence and detailed asymptotic behavior of heteroclinical solutions. In particular, we prove the strict monotonicity of every component, classify the blow-up profiles near the potential wells through Weiss-type monotonicity formulas, and obtain an almost homogeneity property of the solution. Furthermore, after re-parametrizing the heteroclinical solution by its $l_p$-arc length, we derive a curvature estimate for the trajectory near the potential wells. These results provide the geometric control needed for the construction of radial barrier functions for the corresponding Bose-Einstein condensation system.

Computing (adjoint) sensitivities of flows with shocks is a longstanding problem in computational fluid dynamics. The spurious sensitivities due to shock sensors and limiters frequently force practitioners to accept the errors incurred by "freezing" limiters and shock sensors. The recently proposed information geometric regularization (IGR) is an inviscid, PDE-based regularization of the compressible Euler equations that replaces shocks with smooth profiles without damping fine-scale structures. This work derives the forward and adjoint sensitivities for the IGR system with periodic boundary conditions and demonstrates their convergence, under grid refinement, to the sensitivities obtained by finite differences or automatic differentiation through the forward solve.

Let $k$ be a field, $G$ an affine $k$-group scheme, $X$ a connected scheme proper over $k$, $f:(Y,y)\to(X,x)$ a pointed $G$-torsor, $\mathcal{C}_X$ and $\mathcal{C}_Y $ the Tannakian categories of vector bundles on $X$ and $Y$, such that $f^*\mathcal{C}_X\subseteq \mathcal{C}_Y$, we give the necessary and sufficient conditions for the exactness of the natural sequence of $k$-group schemes $1\rightarrow \pi(\mathcal{C}_Y,y) \rightarrow \pi(\mathcal{C}_X,x) \rightarrow G \rightarrow 1.$ This provides a common framework of exact sequences for the $S$-, Nori, $F$-, étale fundamental group schemes and their extended variants. Finally, an isogeny of an elliptic curve shows that the relevant preservation property fails for the local, extended local, and unipotent categories.

The stabilising effect of multiplicative noise for stochastic differential equations, though counterintuitive, has been observed and investigated extensively in last decades. In practice, it is desirable to know if such stabilisation holds also for the discretised setting. In this paper, we address this problem by means of sharp upper and lower estimates for Lyapunov exponents of Milstein schemes for non-autonomous stochastic differential equations. These estimates provide precise large time behaviour in both almost sure and $p$-moment sense. In particular, our results show the preservation of stabilisation from the continuum setting to the discretised setting. One main idea of our analysis is to exploit the second order term concerning the stochastic noise from the Milstein scheme to obtain precise estimates for Taylor expansions of logarithmic and power functions.

The quadratic optimization-free (QO-free) method is a class of powerful and effective algorithms for solving nonlinearly constrained optimization problems in Euclidean spaces. The aim of the present work is to extend this method to solve optimization problems on manifolds with additional equality and inequality constraints. We first present a specific algorithm in the manifold setting. At each iteration, three linear systems sharing a common linear operator are solved to determine the master search direction. In addition, a higher-order correction direction is obtained by solving a reduced linear least squares subproblem to circumvent the Maratos effect which is assumed not to arise in existing related literature. A Riemannian arc search is then performed within the tangent space of the current iterate to generate the new iterate. Under appropriate assumptions, we establish the global and strong convergence of the proposed method. Moreover, we prove that the unit step size will eventually be accepted by the arc search, upon which the superlinear convergence of the algorithm is established. Finally, numerical results demonstrate that the proposed method is very competitive compared with other existing approaches.

In this paper, we use finite fields and linear algebra methods to resolve a conjecture by T. Zhang, L. Zheng, H. Wang, J. Peng and Y. Li (Finite Fields Appl. 110 (2026) 102743) concerning permutation pentanomials.

For an odd prime $p$ and integers $r,q\ge 1$, we study the Butson-type complex Hadamard matrices $H_{p;r,q}(a,b)=\chi(1+p^r ab)$ on $\mathbb{Z}/p^q\mathbb{Z}$, where $\chi$ is a faithful character of the principal-unit quotient. The defining phase is the normalized $p$-adic logarithm, linear modulo $p^q$ exactly when $q\le r$ and nonlinear when $q>r$. We prove that this threshold simultaneously governs monomial equivalence to a finite abelian character table, the cocycle property of the canonical cyclic $2$-cochain, and projective scalarity of the canonical Weyl commutator. We classify the full projective monomial automorphism group as a three-parameter fractional-linear group of order $p^{2q}\varphi(p^q)$, with inverse limit a $\Gamma_0(p^r)$-type congruence subgroup of $\mathrm{PGL}_2(\mathbb{Z}_p)$. For $q>r$, scalar commutators survive on exactly $q-r+1$ maximal valuation-polarized windows; after quotienting radicals, each carries the standard perfect pairing on two cyclic groups of order $p^r$ and the associated finite Heisenberg extension. We determine the exact normalizers of these windows and prove distinct polarizations are pairwise nonconjugate inside the projective monomial group. Finally, abelianization separates $H_{p;r,q}$ from the Fourier matrix $F_{p^q}$ for every $q>r$; low-order defect and fingerprint computations illustrate why this group-theoretic invariant is stronger on the family.

We establish a trapped surface formation theorem for the spherically symmetric Einstein Yang Mills equations in a double-null gauge. The theorem concerns characteristic initial data posed on a pair of transversely intersecting null hypersurfaces and allows nontrivial incoming data. The proof extends the singular characteristic method of An and Lim for the Einstein Maxwell Charged Scalar Field System to the non-abelian Yang Mills setting, where the curvature coupling and gauge field nonlinearities introduce new structural difficulties. This paper constitutes the first part of a program toward weak cosmic censorship for the Einstein Yang Mills system.

Let $K$ be a symmetric convex body in $\mathbb{R}^d$ and let $f(K)$ denote the largest constant $c$ such that every finite collection of translates of $K$ contains a pairwise disjoint subcollection whose total volume is at least $c$ times the volume of the union of the original collection. The classical Vitali covering lemma gives $f(K)\geq3^{-d}$. In this paper, we establish two improvements. First, by a purely combinatorial argument, we prove that $$ f(K)\geq \frac{2}{3^d + 2^d} $$ for every symmetric convex body $K$. This improves the Vitali bound by a factor tending to $2$ as $d$ tends to infinity. Second, using a weighted hard-core model together with a weighted geometric estimate for intersections of Euclidean balls, we show that, for all sufficiently large $d$, $$ f(B^d)\geq \left( \log\frac{3}{1+\sqrt3} -O\left(\frac{\log d}{d}\right) \right)d\,3^{-d}, $$ where $B^d$ is the unit Euclidean ball in $\mathbb{R}^d$. Thus, the classical lower bound is improved by a factor of order $d$.