Papers

This paper establishes sharp upper bounds for $p$-capacities $\mathrm{Cap}_{1<p<\infty}$ in the hyperbolic space $\mathbb{H}^n$ through hyperbolic quermassintegrals and effective curvature radii. The quermassintegral comparisons involve $W_{n-1}$, $W_{k+1}+k(n+1-k)^{-1}W_{k-1}$, and the pair $W_1\mid W_2$. For star-shaped, mean-convex or h-convex hypersurfaces, inverse mean curvature flow further produces curvature radii determined by $L^q$-averages of the normalized mean curvature and by moments of its squared hyperbolic excess. These radii convert the resulting estimates into sharp geodesic-ball comparisons for the capacity-to-area ratio. In the range $p>2m+1$, an interpolating radius combines the $2m$-th curvature-excess radius with the $L^\infty$ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.

Over a decade ago, Koolen, Hayat, and Iqbal posed the problem of whether the distance spectrum determines bipartiteness within the class of connected graphs. In this paper, we resolve this problem in the negative: we explicitly construct an infinite family of counterexamples, where each pair comprises a connected bipartite graph and a connected non-bipartite graph with equal distance spectra.

We demonstrate that, with a suitable rescaling, using Johnson-Lindenstrauss transforms (JLTs) in the random subspace model-based derivative-free optimization (DFO) algorithm from [Cartis & Roberts, Math. Prog. 199 (2023)] achieves an improved first-order worst-case evaluation complexity bound compared to 'unscaled' JLTs. This improved bound does not require any modification to the original algorithm or complexity analysis, and matches the best-known bound in terms of dimension dependence for model-based DFO (e.g. achieved by [Scheinberg & Chaudhry, ICM 2026] using Haar matrices in a similar framework). A similarly refined analysis of the analogous second-order subspace method from [Cartis & Roberts, Opt. Meth. & Softw. (2026)] shows that using suitably scaled JLTs to generate the subspaces gives a better complexity bound, which improves over full-space methods only if the problem has low effective rank.

We develop a column-wise chi-squared geometry for discrete memoryless channels (DMCs) yielding tight, logarithm-free bounds on mutual information, channel dispersion, and finite-blocklength coding rates without evaluating logarithms of the channel matrix. The key parameter is~\(\eta\)—the worst-case relative deviation of a transition probability from its output marginal, which is small precisely when the channel is close to the fully noisy channel $t_{ij}=s_j$. We prove three main results: (1) a third-order ratio expansion showing \(I(X;Y)/\chi^2(X;Y)\to 1/2\) as \(\eta\to 0\) with an \(O(\eta)\) skewness correction; (2) a two-sided dispersion equivalence bounding \(V(X;Y)\) above and below by \(\chi^2(X;Y)\) with explicit constants \(c_{\pm}(\eta)\to 1\); and (3) a certified robust design rate \(R_{\mathrm{cert}}(n,\varepsilon)\) with total certification gap \(O(\eta)+O(\eta/\sqrt{n})+O(\log n/n)\). The certified bounds on \(I\) and \(V\) require only addition, multiplication, division, and square roots; the final rate also uses \(Q^{-1}(\varepsilon)\).

We investigate the behavior of lattice points in 1-symmetric convex bodies–those invariant under both coordinate permutations and sign changes. In this setting we introduce a discrete analogue of the isotropic constant and establish concentration of mass properties for lattice points that parallel classical mass concentration results for convex bodies in isotropic position. These geometric estimates are applied to prove dimension-free $\ell^p$ bounds ($p \ge 2$) for discrete dyadic maximal operators and for maximal functions in the small-scale regime. Furthermore, we demonstrate that coordinate permutation invariance is a necessary condition by constructing 1-unconditional convex bodies in isotropic position for which these dimension-free estimates fail. Our results provide a framework that generalizes several recent works on dimension-free estimates for discrete maximal functions, including those by Bourgain, Mirek, Stein, and Wróbel.

We prove that each $2$-edge-coloured spanning subgraph $G$ of $K_{n,n}$ with $\delta(G)\ge \lfloor (2n+1)/3 \rfloor$ can be covered by at most three monochromatic components. We provide a $2$-edge-coloured spanning subgraph of $K_{n,n}$ showing this minimum degree condition is sharp.

This paper presents an innovative approach to the capacities-to-masses for the Laplace-Beltrami operators on the complete Riemannian manifolds, unexpectedly solving the open problem posed within \cite[Remark 2.16]{GPS} on the limiting variational capacity.

Let $Q$ be a nonzero $s\times t$ $(0,1)$-pattern, and let $m\ge s$ and $n\ge t$. An $m\times n$ matrix is strongly $Q$-forcing if every $1$-entry belongs to an $s\times t$ submatrix equal to $Q$. Let $F^{*}(m,n,Q)$ count these matrices. Put $H=m-s+1$ and $W=n-t+1$. We prove \[ F^{*}(m,n,Q)\ge 2^{HW}. \] Writing $r$ and $c$ for the numbers of nonzero rows and columns of $Q$, equality holds if and only if \[ (H=1\text{ or }r=1)\qquad\text{and}\qquad(W=1\text{ or }c=1). \] Thus the minimum over all nonzero $s\times t$ patterns is $2^{HW}$, attained exactly by singleton patterns when $H,W>1$, and every fixed nonzero pattern has square growth rate $1$. We also refine the count by weight. If $o(Q)$ is the number of $1$-entries of $Q$, then the number of strongly $Q$-forcing matrices at the minimum positive weight $o(Q)$ is $\binom{H+r-1}{r}\binom{W+c-1}{c}$; at every fixed density in $(0,1)$, the logarithmic growth rate is the binary entropy when $m$ and $n$ are comparable. For ordinary forcing, where every $s\times t$ submatrix contains the $1$-entries of $Q$ in their prescribed positions, let $F(m,n,Q)$ be the number of forcing matrices and let $\mathfrak m(m,n,Q)$ be their minimum weight. We prove \[ F(m,n,Q)=2^{mn-\mathfrak m(m,n,Q)} \quad\text{and}\quad 2^{\,mn-\mathfrak m(m,n,Q)+HW} \le F(m,n,Q)F^{*}(m,n,Q) \le 2^{mn}. \] The lower product bound has the same equality cases as the strong-forcing lower bound above, while the upper product bound is attained exactly by singleton patterns. In particular, the product is at least $2$, with equality exactly when $s=m$, $t=n$, and $Q$ is the all-ones pattern.

Maximal Lyapunov functions encode the entire domain of attraction of an asymptotically stable equilibrium, but preserving strict decrease under neural approximation is difficult because its margin vanishes at the equilibrium. For systems locally dominated by an asymptotically stable homogeneous vector field, we construct a continuously differentiable maximal target and an anchored, positivity-preserving neural family. We prove semiglobal universal approximation: strict neural Lyapunov functions and their first derivatives can approximate the target on nested invariant sublevel sets that exhaust the domain of attraction. We also provide directly verifiable conditions under which a candidate neural Lyapunov function can be formally certified, and illustrate the effectiveness of the proposed neural architecture through numerical examples.

We prove an asymptotic structural property of generic initial ideals. This single phenomenon yields results on Hilbert functions, persistence, hyperplane restriction, graded Betti numbers, combinatorial shadow minimization, and Lefschetz properties.

We study a specified Luttinger filling in the construction of Lidman and Piccirillo whose simple connectivity would yield exotic $S^2\times S^2$ and $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. An earlier version of this manuscript claimed simple connectivity from incorrectly derived group relations; that claim is withdrawn. We derive the complete fundamental-group presentation and express the group as a one-relator quotient of an iterated HNN extension containing the genus-two surface group. The filled group is trivial exactly when the final relation kills a specified fiber commutator. We also determine its universal central extension with cyclic kernel, construct geometric realizations of it, and extend the boundary involution over the filling. The final filling problem remains unresolved.

We show that, for Weil–Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H \leq X$ and $H/\log X\to\infty$ as $X\to\infty$. In particular, it applies to intervals which are not necessarily short.

In this paper, we establish a new variant of the Burger-Brooks transfer principle, which allows us to apply spectral estimates for measured Riemannian surfaces to obtain the following result: There exists a universal constant $C>0$ such that, for every connected graph $G=(V, E)$ with boundary $B$, maximum degree $d_{\max}$ and genus $g$, \[\sigma_k(G, B)\leq C d_{\max}\frac{g+k}{|B|},\] where $1\leq k\leq |B|$ and $\sigma_k(G, B)$ denotes the $k$-th Steklov eigenvalue of $G$ with boundary $B$. This bound is sharp up to a universal constant, thereby resolving a problem raised by Lin and Zhao [J. Lond. Math. Soc. (2) 112 (2025), Paper No. e70238]. Furthermore, when $B=V$, the above result yields an upper bound for the Laplacian eigenvalues of graphs, improving the previously known bounds of Kelner, Lee, Price and Teng [Geom. Funct. Anal. 21 (2011), 1117–1143] and Amini and Cohen-Steiner [Comment. Math. Helv. 93 (2018), 203–223].

Polygon equations form a family of equations generalizing the pentagon equation. In this paper, we construct semi-simplicial sets of permitted colorings associated with set-theoretic solutions of polygon equations and use them to define the corresponding (co)homology groups. We investigate several properties of these groups and establish an equivalence of categories between set-theoretic solutions of polygon equations and higher Segal semi-simplicial sets satisfying certain conditions. As a special case, our result recovers the correspondence between bijective set-theoretic solutions of the pentagon equation and $2$-Segal semi-simplicial sets proved by Dyckerhoff–Kapranov.

This paper considers robust nonadaptive regulation for general nonlinear systems in an output-feedback setting with arbitrarily high relative degree. We develop a nonadaptive design that combines an input-driven filter and a generic internal model with a recursive backstepping law, thereby recasting the regulation problem as the robust input-to-state stabilization of an augmented error system. Unlike adaptive schemes, the proposed method does not rely on linearly parameterized regressors and does not require the construction of Lyapunov functions having merely nonpositive derivatives. Under standard assumptions on the exosystem, including purely imaginary and simple eigenvalues, together with a minimum-phase input-to-state stability condition on the internal dynamics, we establish global asymptotic regulation and derive explicit, verifiable inequalities for selecting the design gains. The resulting nonadaptive framework guarantees convergence of the estimation and tracking errors even when the controlled-system dynamics are complex or only partially known. The effectiveness of the theoretical results is demonstrated using a benchmark controlled Duffing system.

We study finite-sample parameter estimation in logistic regression with Gaussian design, where the goal is to estimate $\mathbf{\theta}^*\in \mathbb{R}^d$ with $R=\|\mathbf{\theta}^*\|_2\ge 1$ from i.i.d. samples $\{(\mathbf{x}_i,y_i)\}_{i=1}^n,$ $\mathbf{x}_i \sim N(0,\mathbf{I}_d)$, $y_i\mid \mathbf{x}_i \sim \mathrm{Bernoulli}((1+\exp(-\mathbf{x}_i^\top \mathbf{\theta}^*))^{-1})$. In this paper, we provide the first minimax optimal estimator, and improve on the best known finite-sample error rate for the maximum likelihood estimator (MLE). These two accomplishments are due to a minimax optimal estimator for the parameter norm $R$. First, we establish the minimax lower bound $\Omega(\sqrt{R^3/n})$ for norm estimation. We then improve the best known norm estimation error rate of the MLE, i.e., $O(\sqrt{R^3d/n})$ from Chardon, Lerasle and Mourtada (2024), to $\tilde{O}(\sqrt{R^3/n}+R^2d/n)$. The additional term, $R^2d/n$, appears to be the intrinsic bias of the MLE, as evidenced by the high-dimensional asymptotic theory of Zhao, Sur and Candes (2022) and numerical examples. We show that, however, this additional term is not information-theoretically necessary. To this end, we construct an efficient debiased norm estimator that achieves the error rate $O(\sqrt{R^3/n})$ and is therefore minimax optimal. Combining this with the optimal direction estimator given by the MLE, we establish the minimax optimal rate $\Theta(\sqrt{Rd/n}+\sqrt{R^3/n})$ for estimating $\mathbf{\theta}^*$, as well as the improved finite-sample error rate $\tilde{O}(\sqrt{Rd/n}+\sqrt{R^3/n}+R^2d/n)$ for the MLE. Numerical experiments demonstrate that the proposed minimax optimal estimators outperform the MLE.

First-order methods typically require a specific step-size that depends on the regularity conditions of the objective function, such as the smoothness, Lipschitz continuity, or strong convexity constants. The Polyak step-size is a classical alternative for subgradient descent on convex functions that only uses knowledge of the optimal value of the objective function and automatically adapts to the above-mentioned regimes. However, many optimization problems are better described by non-Euclidean geometries and are more amenable to mirror descent. Extending this adaptivity to mirror descent is subtle. Some existing generalizations of the Polyak step-size rely on norms instead of purely on relative geometry, excluding many of the use cases of mirror descent. In this work, we revisit a variant of the Polyak step-size based on Bregman projections due to Kiwiel (1997), which we call mirror Polyak. This method is known to converge asymptotically, but its convergence rate is not known. We show that mirror Polyak enjoys guarantees similar to its Euclidean counterpart, automatically adapting to relative notions of smoothness, Lipschitz continuity, or strong convexity. We then leverage mirror Polyak to avoid having to know the optimal value in some structured optimization problems such as regularized linear and logistic regression. We propose a lifted formulation based on convex duality with optimal value exactly zero and a natural mirror map given by the problem's structure. Mirror Polyak applied to the lifted problem enjoys the same worst-case guarantees as the Polyak step-size in the original problem if we knew the optimal value.

This paper addresses the limitations of Gaussian distribution assumptions in statistical sparse learning, particularly in modeling correlated and heterogeneous data. Conventional Gaussian models often lack robustness towards outliers and underlying distribution assumptions. To overcome these limitations, we propose the use of the $q$Gaussian distribution, derived from Tsallis entropy maximization, as a robust alternative. This is notably relevant in biostatistics, where the presence of correlated observations and heterogeneity, such as in genetic and longitudinal studies, is prevalent. Our contributions include modeling of correlated data through the re-derived multivariate probability density function from Tsallis entropy maximization, thereby addressing the limitations inherent in conventional Gaussian models. Furthermore, we introduce a novel framework that adapts numerical methods designed to find equilibria in flows to tackle composite optimization problems prevalent in statistical sparse learning. Applying this framework to the Hager-Zhang conjugate gradient algorithm \cite{Hager2005}, we develop a numerically stable and efficient algorithm for sparse statistical learning. The $q$Gaussian distribution, informed by the principle of maximizing Tsallis entropy, presents a viable and flexible alternative to Gaussian-based methods. This paper not only contributes to the theoretical understanding of statistical distributions and optimization techniques, but also paves the way for practical data analysis.

The modular Mandelbrot set $M_\Gamma$, the connectedness locus of the modular mating family of 2 : 2 holomorphic correspondences $\mathcal{F}_a$ on the Riemann sphere, is homeomorphic to the classical Mandelbrot set $M$. The Klein combination locus $\mathcal{K}$ (the "discreteness locus" of the family $\mathcal{F}_a$) is a pinched neighborhood of $M_\Gamma$ in the $a$-plane, pinched at the root point. We construct a canonical map $\Psi$ from $\mathcal{K}\setminus M_\Gamma$ into the hyperbolic plane $\mathbb{H}$, inspired by the construction of Douady and Hubbard for their celebrated conformal bijection $\Phi: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$, and we prove that $\Psi$ is analytic. This map $\Psi$ induces a tessellation of $\mathcal{K}\setminus M_\Gamma$ by pulling back a tessellation of $\mathbb{H}$ invariant under the modular group. We develop a series of conjectures concerning the structure of $\mathcal{K}$, its boundary, and $\Psi(\mathcal{K}) \subset \mathbb{H}$.

In this work, we introduce the Koszul solitons of the Hesse-Koszul flow, i.e., self-similar solutions for a geometric flow on Hessian manifolds. We obtain some partial differential equations and basic properties of Koszul solitons, and provide some examples. The key idea is to investigate the relationship between affine manifolds and complex manifolds, this method can also be used to study some other geometric problems on Hessian manifolds. It is worth noting that Hessian manifolds, as a special case of affine manifolds, have wide applications in statistics and information geometry.

Let $F_m=F(a_1,\dots,a_m)$ with $m\ge 2$, and fix $q\ge 1$. For every fixed $0<\lambda<1$, we prove that a $q$-tuple $\mathbf W_n$ of independent uniformly random cyclically reduced words of length $n$ is \emph{$\lambda$-stable} with probability converging to $1$ exponentially fast. Namely, for every $\Phi\in Aut(F_m)$, the tuple $\Phi(\mathbf W_n)$, after cyclic reduction and symmetrization, satisfies the $C'(\lambda)$ small cancellation condition. Combining generic $\lambda$-stability with Greendlinger normal-closure rigidity and with previous results of Kapovich-Schupp-Shpilrain on generic Nielsen uniqueness and generic Whitehead rigidity we establish, for any fixed $m\ge 2, q\ge 1$, isomorphism rigidity for generic $m$-generator $q$-relator groups. Thus we show that two such generic groups $\langle a_1,\dots, a_m| r_1,\dots, r_q\rangle$ and $\langle a_1,\dots, a_m| s_1,\dots, s_q\rangle$ are isomorphic if and only if, after possibly permuting and inverting the generators $a_1,\dots, a_m$, the relator tuples $(r_1,\dots, r_q)$ and $(s_1,\dots, s_q)$ are the same, up to reordering, cyclic permutations and inverting the relators. Among the applications, we obtain a quadratic-time algorithm that generically solves the isomorphism problem for $m$-generator $q$-relator groups, and show that the number of isomorphism types represented by $m$-generator $q$-relator presentations with cyclically reduced relators of length $n$ is asymptotic to \[ \frac{(2m-1)^{qn}}{2^{m+q}m!\,q!\,n^q}. \] The proof of generic $\lambda$-stability relies on the use of geodesic currents and on our deterministic sufficient criterion for a $q$-tuple $\mathbb W$ in $F_m$ to be $\lambda$-stable in terms of the components of $\mathbb W$ being sufficiently projectively close to filling currents.

We prove that any two smooth toric Fano fourfolds whose integral cohomology rings are isomorphic as graded rings are diffeomorphic.

Let $K$ be an origin-symmetric full-dimensional convex polytope in $\mathbb R^n$, $n\ge 3$. We show that, for $-n+3<p<1$, the normed space $(\mathbb R^n,\|\cdot\|_K)$ embeds in $L_p$ if and only if it embeds in $L_1$. The case of $p\le 0$ is understood in the generalized sense. In particular, an origin-symmetric full-dimensional convex polytope $K\subset\mathbb R^n$, $n\ge 5$, is an intersection body if and only if $K$ is the polar of a zonotope.

Let $K_{r,s,t}$, with $r\le s\le t$, denote the complete tripartite graph whose partite sets have sizes $r,s,t$. Mahmoodian and Mirzakhani gave three necessary conditions for $K_{r,s,t}$ to admit a decomposition into 5-cycles and conjectured that these conditions are sufficient. One of the conditions is $t\le 4rs/(r+s)$. We prove the conjecture for every odd triple on the extremal boundary $t = 4rs/(r+s)$. The proof is constructive. After reducing an arbitrary odd boundary triple to $(r,s,t)=(hga,hgb,hab)$, $a+b=4g$, we give an explicit cyclic decomposition of $K_{ga,gb,ab}$ and use the Mahmoodian and Mirzakhani scaling theorem to supply the common factor $h$. Together with the previously known all-even result, this settles the conjecture for every triple satisfying the boundary condition with equality. We also report explicit computer-generated $C_5$-decompositions for 117 odd triples satisfying the necessary conditions, 116 of which are strict-interior cases. To the best of our knowledge, all 117 cases were previously unresolved: no decomposition for any of them had been reported, and none of the 117 triples is covered by earlier existence results, constructions, or their recursive consequences. Moreover, these 117 certificates together with the boundary construction settle every previously unresolved triple satisfying the necessary conditions with fewer than $4400$ edges. Each computation is supplied as a machine-readable cycle-list certificate and can be checked independently by a short Python verifier. We also give a complete human-readable edge-label-matrix certificate for $K_{9,19,23}$.

We explore the problem of asymptotically stabilizing a class of interacting particle systems to a prescribed particle configuration with zero velocity. The class of interacting particle systems is motivated by the Cucker-Smale model on the mean-field level. Rather than designing a feedback by manually solving an IDA-PBC matching equation, we derive a feedback using an instantaneous rolling horizon control algorithm. We show that this feedback preserves the port-Hamiltonian structure of the interacting particle system and therefore allows for an interpretation in the IDA-PBC framework. Additionally, the energy balance of the closed-loop Hamiltonian provides an important tool to prove convergence of the particles' velocities and interaction forces in discrete and continuous time. If the control operator is surjective, we additionally prove convergence of the particle positions to the prescribed configuration.

We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters $\beta_K=-K$ and $\alpha_K=c/(K+1)$. We write out the parameter preserving first order ladder relations at fixed $(\alpha,\beta)$, with real constants. Combining them returns the three-term recurrence; iterating them builds $R_n^{\alpha,\beta}$ from the constant polynomial. We solve in triangular form the connection problem with the $\alpha=0$ family, identified with the Gegenbauer polynomials, and transfer the ladder relations to the $(n,\ell)$ lattice. Under $x=\cot\chi$ the in-level operators depend on $\ell$ alone. The dressed functions carry intrinsic lowest weight $\mathrm{su}(1,1)$ modules along columns of fixed $\ell$; rows need a boundary prescription and a rescaling. The deformed functions carry the same modules through higher order operators, which yield relations between Romanovski polynomials with different parameter pairs. Algebraically the family lies in the Jacobi class, and we indicate throughout which statements are inherited from that framework by transformation or specialization and which are not.

Let $\binom{[n]}{k}$ be the set of all $k$-element subsets of the set $\{1,\ldots,n\}$ and let $\mathcal A,\mathcal B \subseteq \binom{[n]}{k}$ be two cross-intersecting families, that is, $A\cap B\neq \emptyset$ for any $A\in \mathcal A$ and $B\in \mathcal B$. The classical cross-intersecting version of the Erdős–Ko–Rado theorem, due to Pyber and Matsumoto–Tokushige, states that if $n\geq 2k$, then $|\mathcal A||\mathcal B|\leq \binom{n-1}{k-1}^2,$ where the equality holds for $n>2k$ if and only if $\mathcal A=\mathcal B$ is a star. In the present paper, we first give a stability result of this theorem by using Filmus's FKN theorem on the slice and linear algebra method as follows: There exists a constant $C>1$ such that if $n\geq 2.07k$ and $|\mathcal A||\mathcal B|\geq (1-\epsilon)\binom{n-1}{k-1}^2$, where $\epsilon\leq \frac{k^2}{C^2 n ^2 }$, then there is a star $\mathcal{S}$ such that $|\mathcal{S} \Delta \mathcal A|\leq C \epsilon \binom{n}{k}$ and $|\mathcal{S} \Delta \mathcal B|\leq C \epsilon \binom{n}{k}.$ Moreover, based on this stability result and the eigenvalues of the matrices of the Johnson scheme, we present an Erdős–Ko–Rado theorem for cross-intersecting families in the Euclidean inner product showing that if $n\geq 2k$ and $k\geq d \geq 0$, then $$\big\langle\mathbf{v}_d(\mathcal A),\mathbf{v}_d(\mathcal B)\big\rangle \leq \frac{\binom{k}{d}\binom{k-1}{d}}{\binom{n-1}{d}}\binom{n-1}{k-1}^2 +\binom{k-1}{d-1} \binom{n-d-1}{k-d}\binom{n-1}{k-1},$$ together with uniqueness and a corresponding stability result, where $\mathbf{v}_d(\mathcal A) \in \mathbb R^{\binom{[n]}{d}}$ is the $d$-degree vector of $\mathcal A$ whose $U$-entry is the number of members in $\mathcal A$ containing $U$.

Following previous work on metric bisectors with respect to the Korányi metric, we study here bisectors in the Heisenberg group $\mathfrak{H}$ equipped with the Carnot–Carathéodory metric $d_{cc}$.

In this paper, we introduce polynomial $\mathcal{Z}$-contraction by unifying the ideas of polynomial contraction and $\mathcal{Z}$-contraction and also provide demonstrative examples to show the effectiveness and novelty. Chiefly, we prove two fixed point results for polynomial $\mathcal{Z}$-contraction employing simulation functions. Lastly, we utilize our main result to discuss the existence and uniqueness of solution for boundary value problem with a concrete example in $L^p(0,\pi)$ under suitable assumptions.

It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford $C^{r-1}Z$ gate, for every constant $r\geq 3$. In particular, we obtain length-$n$ quantum LDPC codes (with constant-weight stabilizers) of polynomial distance $d\geq n^{(1-\epsilon)/r}$ supporting transversal $C^{r-1}Z$ gates on a close-to-linear number $k\geq n^{1-\epsilon}$ of disjoint tuples of logical qubits, for arbitrarily small $\epsilon>0$. Our construction is the first with constant-weight stabilizers that obtains $dk\gg n$, and as a consequence achieves arbitrarily small magic state overhead exponent $\gamma=\log(n/k)/\log(d)>0$. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many $k=\Omega(n)$ logical $C^{r-1}Z$ gates, though with stabilizer weight and physical circuit depth $n^\epsilon$. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal $C^{r-1}Z$. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.