Papers

We study multipartite hypergraphs definable in $1$-semi-equational theories and prove almost-linear Zarankiewicz bounds in every fixed arity $r\geq2$. If $T$ is a $1$-semi-equational theory, then, for every formula $\varphi$ and fixed $t,r\geq2$, there is a constant $c$ such that each $K_{t,\ldots,t}$-free $r$-partite hypergraph defined by $\varphi$ on $n$ vertices has $O_{T,\varphi,t,r}\!\left( n^{r-1}(1+\log(1+n))^c \right) $ edges. Put $\alpha_k=\min\{k-1,2\}$. In the bipartite case, a Boolean combination of $m$ $(k,1)$-semi-equations has $ O_{k,t,m}\!\left( n(1+\log(1+n))^{(m-1)\alpha_k} \right) $ edges whenever it is $K_{t,t}$-free. In particular, a relation defined by one $(k,1)$-semi-equation or its negation has a linear bound. The proofs combine incidence estimates for indexed set systems with low-crossing orderings of finite $k$-wise laminar families. Consequently, no $1$-semi-equational theory locally trace-defines an infinite domain.

We establish non-asymptotic Berry–Esseen bounds and bootstrap approximations for the Hilbert-space norm of nondegenerate $U$-statistics. Our triangular-array framework allows the kernel to take values in a separable Hilbert space $H_n$ that may vary with the sample size, thereby covering both infinite-dimensional spaces and Euclidean spaces of increasing dimension. The Gaussian approximation combines a Hilbert-space version of Stein's method with refined control of the higher-order terms in the Hoeffding decomposition. Its error depends on fourth moments of the kernel and on the spectral geometry of the covariance operator of the Hájek projection, particularly the proportion of squared spectral mass lying outside the leading eigendirection. Consequently, the theory accommodates singular and approximately low-rank covariance operators, provided that sufficient spectral mass remains beyond the leading direction. For $H_n=\mathbb{R}^{d_n}$, we obtain dimension-explicit bounds under coordinatewise fourth-moment conditions. We also establish approximation bounds for the empirical, Gaussian-weighted, and jackknife multiplier bootstraps, yielding tests with asymptotically correct size and consistency against alternatives at covariance-dependent separation rates. For testing the vector of pairwise Kendall's tau coefficients, a matching minimax lower bound shows that the resulting separation rate is minimax rate-optimal over dense Gaussian correlation alternatives.

Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the Hölder conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\geq5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\geq6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\leq p<4/3$ and an $\ell^{4/3}$-to-$\ell^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman–Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.

In this paper, we focus on left-compressed families \(\mathcal{F}(\mathcal{G})\) generated by a family of sets \(\mathcal{G}\). We establish a necessary and sufficient condition on \(\mathcal{G}\) under which \(\mathcal{F}(\mathcal{G})\) is intersecting. We then introduce a construction of maximal left-compressed intersecting families \(\mathcal{F}_G\) from a single generator by adding suitable sets not containing \(1\) to \(F(G)\). From this, we derive several estimates for the size of \(\mathcal{F}_G\).

We introduce two new arithmetic invariants associated with Hermitian line bundles on arithmetic varieties, which play the role of the Bergman distortion function in the arithmetic setting. As a first step, we show that these two functions are asymptotically equivalent. This leads to the introduction of an intrinsic measure $ \widehat{\mu}^{\sup}_{\mathrm{eq}},$ which we show to be a natural arithmetic analogue of the equilibrium measure. We prove that $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ serves as a detector of arithmetic positivity. In particular, we determine it for weakly nef Hermitian line bundles and, in full generality, in the toric case. Moreover, $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ governs the arithmetic volume function and its variational properties. More precisely, we establish weak integral representations of the arithmetic volume and of its first variation in terms of $\widehat{\mu}^{\sup}_{\mathrm{eq}}$. These formulas provide a measure-theoretic interpretation of the volume derivative of Yuan–Zhang and reveal a structural parallel with variational formulas in complex pluripotential theory. Our approach differs fundamentally from previous treatments based on Harder–Narasimhan filtrations or arithmetic Okounkov bodies. It places the arithmetic volume at the center of the theory and suggests that asymptotic volume invariants, rather than heights, provide the natural analytic framework for problems in equidistribution and arithmetic dynamics.

Ellingsrud, Göttsche, and Lehn proved a remarkable universality result for tautological integrals on Hilbert schemes of points on surfaces. We prove a relative form: over a base $B$, the tautological pushforwards are governed by universal power series paired with relative $\kappa$-classes. As applications, we focus on Segre classes on families of curves and surfaces. For curve families, we determine all universal coefficients for total Segre pushforwards. These formulas answer a question of Oprea–Pandharipande, and the resulting recursions are surprisingly related to monotone Hurwitz numbers. For surface families, motivated by Marian–Oprea–Pandharipande's work on Lehn's conjecture, we obtain a codimension-one relative form of the conjecture, with explicit formulas for every universal series.

Phylogenetic trees are used in evolutionary biology to represent the evolutionary history of a collection of taxa. As we have incomplete information about any evolutionary history, recovering trees from partial information is a focus of phylogenetic combinatorics. However, in some cases the available data does not describe a single phylogenetic tree. We consider recovery of pairs of phylogenetic trees from their combined subtrees with $k$ leaves, which we call a $k$-bucket. We establish the exact cases in which these pairs of trees are recoverable from their subtrees with a single leaf removed, both when just considering the structure of the trees, and when additionally considering the set of taxa on the leaves. We also consider recovery of pairs of trees with labelled leaves from their rooted triples, and establish that they are recoverable up to a sequence of subtree swaps.

We study quasi-Galois points, which are a generalization of Galois points. We characterize smooth quartic surfaces admitting a quasi-Galois point in terms of K3 surfaces with a certain involution, and provide a criterion for quasi-Galois points for quartic surfaces. Moreover, we also determine all quasi-Galois points of the Fermat quartic surface and show that it has exactly 28 quasi-Galois points.

We study axisymmetric self-similar solutions of the stationary Navier–Stokes equations with $(-3)$-homogeneous forces and no-slip boundary conditions in a half-space. For a fixed force without swirl, we characterize the interval of amplitudes for which solutions exist and describe the velocity singularity at its finite endpoints. For each parameter in this interval, a solution exists after adding a swirl force of any size, and it is unique among solutions with bounded profiles when the swirl force is sufficiently small. The proof uses the nonnegative contribution of swirl to a Riccati equation and a comparison estimate. We also prove that, when the tangential curl of the force is sufficiently small, there exists a unique solution with small velocity profiles.

Let $K$ be a $p$-adic field. We study the arithmetic theory of abeloid varieties over $K$. Our aims are twofold. First, we study the theory of rigid analytic 1-motives, which will be viewed as a tool to describe degeneration of abeloid varieties, similarly as in the classical algebraic setting. Our key new results are the equivalence between formal (resp. log formal) 1-motives over $\mathcal{O}_K$ and rigid analytic 1-motives with good (resp. semi-stable) reduction over $K$, and the Néron-Ogg-Shafarevich criterion for the good (resp. semi-stable) reduction of rigid analytic 1-motives. In particular, we construct log formal 1-motives and log $p$-divisible groups over $\mathcal{O}_K$ from semi-stable abeloid varieties over $K$. Next, we study the conjugate uniformization of an arbitrary abeloid variety $A$ over $K$. This is a type of $p$-adic uniformization initiated by Iovita–Morrow–Zaharescu in case of abelian varieties with good reduction. Our approach here is based on Fargues' theory of $p$-divisible rigid analytic groups. In fact, we view the theory of conjugate uniformization as a study of rational points of dualizable $p$-divisible rigid analytic groups in terms of their classification Hodge–Tate triples. Along the way, we construct $p$-divisible rigid analytic groups from rigid analytic 1-motives.

In this paper, an ontology-driven semantic communication (ODSC) framework is proposed for multi-view visual question answering (VQA) tasks. In the considered framework, multiple transmitters observe a scene, extract the semantic information (SI) with vision-language models (VLMs), and transmit the scene graphs to a receiver. Due to the completeness, heterogeneity, and uninterpretability of the VLMs, the extracted scene graphs are redundant, ambiguous, and inconsistent. To solve these problems, the transmitters and the receiver share an ontology-based knowledge base that predefines synonyms, inference rules, and consistency constraints. For each transmitter, the proposed ODSC framework removes the partial scene graph that can be inferred based on the inference rules. For the receiver, the proposed framework aligns the SI of different views based on the synonyms and detects the inconsistency among the views based on the constraints. A metric of multi-view VQA accuracy (MVA) is defined to evaluate the proposed framework. Simulation results show that, compared with transmitting the complete scene graphs, the proposed framework reduces the data size of the SI by up to 87.1% while improving the answering accuracy by 4.5%. Moreover, the proposed framework yields up to a 16.0% improvement in terms of the MVA compared with the SI filtering approaches.

The Stochastic Simulation Algorithm (SSA), widely considered an exact algorithm for stochastic chemical reaction networks, suffers from high computational cost. In this work, we propose a data-driven effective model that operates on a user-defined coarse time step independent of the underlying microscopic reaction-event scale. This is accomplished by directly approximating the finite-time transition kernel of the continuous-time Markov chain induced by SSA, using a generative machine learning model trained on short bursts of SSA simulation data. The trained model constructs a stochastic propagator that recursively generates statistically consistent trajectories at the constant coarse time step, with significantly reduced computational cost. In this paper, we employ conditional normalizing flow as the stochastic propagator. A comprehensive set of numerical examples is presented to demonstrate the accuracy and efficiency of the proposed method.

In this paper, we construct the explicit characteristics for the $\lambda$-dissipative solutions ($\lambda\in[0,1]$) to the two-component Hunter–Saxton (2HS) system. Using these characteristics, we provide a comprehensive study of the existence, uniqueness, and asymptotic behavior of these solutions. For the fully dissipative case ($\lambda=1$), uniqueness follows from the absence of outgoing cusps. In the partially dissipative regime ($0<\lambda<1$), outgoing cusps are present. We formulate an exact Eulerian dissipation rule which identifies the energy that has already passed through wave breaking by the intrinsic condition $\rho=0$ and $u_x\geq 2/t$. This rule determines the dissipated part of the energy measure and yields uniqueness. This uniqueness applies to the classical Hunter-Saxton equation and seems to be the first uniqueness result for the general $\lambda$-dissipative solutions. Concerning the large-time dynamics, we show that the density $\rho$ and the singular part of the energy measure decay to zero as $t\to\infty$, indicating that all energy is eventually concentrated in the $u$-component. Moreover, we derive the leading-order asymptotic term, which takes the form of a kink-wave determined by the system's remaining energy. This kink-wave can be explicitly computed from the initial data and the dissipation parameter $\lambda$.

The game of Cops and Robbers is a classical pursuit–evasion game on graphs. For a graph $G$, the cop number $c(G)$ is the minimum number of cops needed to guarantee the capture of a robber on $G$. Although this parameter has been determined for several fundamental graph classes, comparatively few exact results are known for partial cubes and their subclasses. We first establish an upper bound for every finite median graph $M$ in terms of its tree-dimension, which improves Crawford and Iršič Chenoweth's bound significantly. This result refines the previous upper bound expressed in terms of a hypercube embedding dimension and can give a substantially smaller estimate. Then we investigate the cop numbers of simplex graphs—a subclass of partial cubes. For a finite graph $G$, the simplex graph $S(G)$ has the cliques of $G$, including the empty clique, as its vertices, with two cliques adjacent whenever they differ in exactly one vertex. We establish a general lower bound for $c(S(G))$ in terms of the clique number of $G$ and a general upper bound in terms of its chromatic number. Finally, as direct applications, we determine the exact values of cop numbers of some special simplex graphs—bipartite wheels, Fibonacci and Lucas cubes.

In [Elkies 2006, Elkies 2007] we announced a K3 surface X/Q with Néron–Severi group NS(X) = NS_Q(X) of rank 19 and an elliptic fibration of Mordell–Weil rank 17. This is the largest possible Mordell–Weil rank over Q(t) for an elliptic K3 surface. One of the fibers of this surface is the elliptic curve of rank at least 28 that was the elliptic curve of highest rank known during the years 2006–2024. We further announced that there are quadratic base changes of this fibration to elliptic surfaces of Mordell–Weil rank 18 over Q(t), and that there are pairs of such quadratic base changes whose compositum is an elliptic surface of Mordell–Weil rank 19 over an elliptic curve E_0/Q with #(E_0(Q)) = \infty, whence there are infinitely many elliptic curves of rank at least 19 over Q. We exhibit one such pair. A subsequent paper will show (using a technique also announced in [Elkies 2007] how we computed X and its fibration.

The blowup equations of Huang, Sun and Wang are bilinear relations satisfied by the refined topological string partition function of a local Calabi–Yau 3-fold. We prove the blowup equations for the local Hirzebruch surfaces \(\operatorname{Tot}_{\mathbb F_\ell}K_{\mathbb F_\ell}\), \(0\leq\ell\leq2\), as identities of torus-equivariant symmetrized \(K\)-theoretic stable pair invariants; for \(\ell=2\) the invariants are localized indices. The proof identifies the stable pair vertex sum, after division by the fibre contribution, with the equivariant Euler characteristic of \((\det\mathcal V)^\ell\) on the moduli space of framed rank \(2\) sheaves on \(\mathbb P^2\). The blowup formulas of Nakajima–Yoshioka for framed sheaves then yield the unity and vanishing equations. For local \(\mathbb P^2\) we obtain the blowup equations conditionally on two explicitly stated conjectures. We also state the Huang–Sun–Wang conjecture for general local Calabi–Yau 3-folds in the language of stable pairs, and formulate stable pair conjectures for the local rational elliptic surface involving the \(E_8\) lattice.

This paper is devoted to studying the strong solutions to the vacuum free boundary problem for the isentropic compressible Navier-Stokes equations with density-dependent viscosities under spherical symmetry, which models the motions of isentropic compressible viscous flows surrounded by vacuum. We construct a class of expanding global solutions when the initial data is a small perturbation of the expanding affine solutions for the adiabatic exponent $\gamma>1$ and the viscosity coefficients proportional to $\rho^{\gamma}$, with $\rho$ being the fluid density. In addition, when the viscosity coefficients tend to zero, the perturbed solutions are proved to converge to the solutions to the compressible Euler equations with an explicit converging rate of viscosity coefficients.

Let $(M^{n+1},g)$ be a compact Riemannian manifold with boundary. Under the assumptions $\Ric_g\geq ng$ and $\II_g\geq0$, Wang proposed a sharp strengthening of the Choi–Wang–Reilly estimate, asserting that the first nonzero Laplace eigenvalue of the boundary is at least $n$; see [J. Geom. Anal. 31 (2021)]. We disprove this assertion in every dimension $n+1\geq3$. More precisely, we construct a sequence of metrics on the hemisphere $\Sph^{n+1}_{+}$ converging in $C^\infty$ to the round metric and satisfying \[ \Ric_g>n g,\qquad \II_g>0,\qquad \lambda_1(\partial\Sph^{n+1}_{+},g|_{\partial\Sph^{n+1}_{+}})<n. \] The construction starts from Zhu's infinitesimal conformal deformation, which lowers one branch of the first boundary eigenspace while preserving the normalized Ricci lower bound to first order. We add a multiple of the spherical height function.

We develop a $\mathbb{Q}_p$-homotopy theory for $p$-complete spaces. To a $p$-complete space $X$, we associate a commutative differential graded algebra over $\mathbb{Q}_p$ by rectifying the $E_\infty$-algebra $S^*(X;\widehat{\mathbb{Z}}_p)\otimes_{\widehat{\mathbb{Z}}_p} \mathbb{Q}_p$ of singular cochains. For nilpotent $p$-complete finite type spaces, we prove that the minimal model of this algebra recovers the $\mathbb{Q}_p$-homotopy groups and Whitehead products, in direct analogy with Sullivan's rational homotopy theory. We also prove that, for a non-simply-connected $p$-complete space, the Lie algebra dual to its $1$-minimal model is the Lie algebra of the continuous Mal'cev $\mathbb{Q}_p$-completion of the fundamental group. We apply the $\mathbb{Q}_p$-homotopy theory to several questions in topology and algebraic geometry, including finite realization problems for $p$-complete spaces, finiteness properties of étale homotopy types, formality of smooth proper varieties, Galois representations on étale homotopy groups, and constraints on étale fundamental groups.

We prove the 3-Decomposition Conjecture: every finite connected cubic loopless multigraph decomposes into a spanning tree, a 2-regular subgraph, and a matching. The proof rests on a new theorem on matching complements in subcubic graphs. Let H be a finite connected bridgeless simple graph of maximum degree three, and let S be its set of degree-two vertices, with |S| = k >= 2. We show that H has a matching of size equal to its cyclomatic number, |E(H)| - |V(H)| + 1, whose deletion leaves a single tree containing all of S, together with cycles disjoint from S. The proof is by induction on k.

Let $\kk$ be a \emph{field of characteristic $p>0$, not assumed algebraically closed}, and let $V=\kk[x]/(x^p)$ be the Novikov algebra with product $a\circ b=ab'$. For $\lambda\in\kk$, let $M(\lambda)$ be Xu's module. We compute the second cohomology $\Ht(V,M(\lambda))$ for all $\lambda$ and $p$, and describe the associated abelian extensions. The computation utilizes a $\Zp$-graded presentation $V\cong\kk[t]/(t^p-1)$ to bypass truncation issues and simplify cocycle identities. We determine the exact dimensions of $\Ht(V,M(\lambda))$, showing it is $0$ for $\lambda\notin\Fp$, $3$ for $\lambda\in\Fp$ with odd $p$, and $4$ for $\lambda\in\Fp$ with $p=2$. Explicit cocycle representatives are provided for all cases. As corollaries, we show that every abelian extension of $V$ by $M(\lambda)$ splits when $\lambda\notin\Fp$. We also treat the characteristic-$0$ analogue $P=\kk[t]$: Xu's parameter $\lambda$ collapses to the single value $\lambda=0$, and $\Ht(P,M(\lambda))=0$ throughout, so $P$ is rigid (in fact formally rigid) while its positive-characteristic truncation never is — within this family, it is truncation rather than positive characteristic per se that destroys rigidity.

This paper summarises our previous work on a fully $3$D Summation-by-Parts scheme, derived for a class of linear wave equations on hyperboloidal slices on a fixed Minkowski background. The scheme is derived in spherical polar coordinates, and allows having grid points at the origin and on the $z$-axis, despite coordinate singularities, and at infinity, by introducing compactification followed by rescaling, and is proved to be stable. Reducing it to the standard Cauchy problem, or to finite spacelike slices with an outer boundary, will follow a similarly. Second-order accurate finite-difference methods are used to implement this scheme numerically, but higher-order finite-difference or spectral methods could also be used. Kreiss-Oliger dissipation operators are generalized to curvilinear coordinates and are defined everywhere in the domain, including at the boundary points, such that they satisfy the dissipative property in the energy norms. We also propose new norm convergence tests that include all the grid points at all resolutions and produce more accurate results. Promising results are obtained, giving hope for application to fully nonlinear systems, like the Einstein Field Equations, and extracting the resulting gravitational waves free of systematic errors or gauge ambiguities.

We study the Sherrington-Kirkpatrick model with an additional Curie-Weiss ferromagnetic interaction (SKFI), whose phase diagram in the plane of the disorder strength $\beta$ and the ferromagnetic coupling $\gamma$ consists of a paramagnetic, a ferromagnetic, and a spin glass region. Our main result is a central limit theorem for the free energy in the ferromagnetic regime: at scale $N^{-1/2}$, the fluctuations coincide with those of the Sherrington-Kirkpatrick model in an effective external field determined by the limiting magnetization. We prove this for general mixed even $p$-spin interactions, using the fluctuation theory of Chen, Dey, and Panchenko. For the pure 2-spin model we complete the picture of the phase diagram: we give a new proof of the order-$N^{-1}$ Gaussian fluctuations in the paramagnetic regime, obtained earlier by Banerjee, valid up to the critical window; we determine the limiting distribution on the critical line $\gamma = 1 + \theta N^{-1/2}$ separating the paramagnetic and ferromagnetic regimes; and we show that in the spin glass regime the fluctuations coincide with those of the zero-field SK model under a widely believed variance-divergence assumption. Our results form the Ising analogue of the fluctuation results of Baik and Lee for the spherical SKFI model.

Let $X$ be a smooth projective curve and let $G$ be a reductive group over an algebraically closed field of characteristic zero. We compute the image of arbitrary finite-rank local systems on $\mathrm{Bun}_G(X)$ under the geometric Langlands equivalence, confirming a conjecture of V. Lafforgue in the case of the constant sheaf.

Generalized Grassmannians form a fundamental class of flag manifolds associated with Lie groups. The purpose of this paper is to classify self-maps of generalized Grassmannians of nonzero degree in terms of their induced actions on cohomology. We prove a rigidity theorem showing that, despite the rich and intricate structure of their cohomology rings, the induced cohomology endomorphisms fall into only two natural types: Adams operations determined by the degree and Dynkin symmetries arising from automorphisms of the Dynkin diagram. Combining the geometry of root and weight systems, the actions of Weyl and Dynkin symmetries on Schubert classes, and Bott–Samelson desingularizations of Schubert varieties, our approach applies uniformly to generalized Grassmannians of all Lie types.

The set of $\tau$-approximable numbers, $\mathcal W(\tau)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad^\gamma$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $\Psi$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $\psi$, we prove that $\mathcal W(\psi)\cap\Bad^\gamma$ is strong $\Psi$-rapid winning for a natural gauge $\Psi$ determined by $\psi$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $\Psi$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $\Psi$. In particular, for $\psi(q)=q^{-\tau}, \tau\ge1,$ we recover the exact Jarník–Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(\tau)\cap\Bad^\gamma\bigr)=\frac{2}{\tau+1}.$$

A typical example of a quandle is the conjugation quandle. A quandle is said to be admissible if it is isomorphic to a conjugation quandle. We study the admissibility problem for quandles, that is, determining whether a given quandle is admissible. In particular, we focus on generalized Alexander quandles, which are groups equipped with quandle structures defined by group automorphisms. In this paper, we provide a sufficient condition for admissibility in terms of geometric properties of quandles: namely, if every antipodal set in each algebraically connected component consists of a single point, then the quandle is admissible. This result generalizes previous results on generalized Alexander quandles.

We prove Liouville-type theorems for smooth solutions $(u,p)$ of the stationary Navier–Stokes equations in $\mathbb R^3$ satisfying the finite Dirichlet energy condition and the uniform decay condition. Our results give geometric refinements of two recent Osgood-type criteria, namely the relative decay criterion and the weighted integrability criterion formulated in terms of the head pressure $Q=\frac{1}{2}|u|^2+p$. The proof exploits several properties of the head pressure and the scalar triple-product structure of $(u\times\omega)\cdot\nabla Q$. Combined with an Osgood-type representation of the total vorticity energy, this geometric structure yields Liouville-type criteria involving only the tangential interaction of $u$ and $\omega$ over the superlevel sets of $|Q|$. In the relative decay setting, they lead to Osgood-scale smallness conditions that allow subcritical growth beyond the corresponding uniform bounds. In the weighted integrability setting, the previous conditions involving the full velocity and velocity gradient are refined to involve only the tangential components of $u$ and $\omega$ along the level surfaces of $Q$, together with an angular factor measuring their relative orientation.

The thin set theorem $\mathsf{RT}_{<\infty,\ell}^{n}$ asserts that for every natural number $k$, each coloring $c\colon[\mathbb{N}]^n \to \{0,1,\dots,k-1\}$ admits an infinite set $H$ such that $|c([H]^n)| \le \ell$. Within the framework of the reverse mathematics of second-order arithmetic, $\mathsf{RT}_{<\infty,\ell}^{n}$ implies the $\Sigma_{n+1}^{0}$-bounding principle ($\mathsf{B}\Sigma_{n+1}^{0}$) over $\mathsf{RCA}_0$ for all natural numbers $n, \ell \ge 1$.

We determine the Hausdorff dimension of the intersection of the set $\mathcal W_3(\lambda)$ of simultaneously $\lambda$-well approximable points with the Veronese curve $\mathcal V_3 \subset \mathbb R^3$ for $\lambda \ge3/5$, thus completing the full range of $\lambda$ values. Precisely, we show that for $\lambda\ge 1/3$, $$ \dim\bigl(\mathcal W_3(\lambda)\cap\mathcal V_3\bigr)= \max\left\{\frac{2-2\lambda}{1+\lambda}, \frac{2}{3(1+\lambda)}\right\}. $$ To the best of the authors' knowledge, this makes $\mathcal V_3$ the first nondegenerate, non-planar curve with a completely determined Hausdorff dimension theory.