Papers

The Brown–Erdős–Sós problem is a fundamental problem in sparse hypergraph Turán theory. For integers $r,k\ge 2$ and $s\ge r$, the problem asks for the maximum number $f^{(r)}(n;s,k)$ of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. In 1971, Brown, Erdős, and Sós proved that $f^{(r)}\bigl(n;(r-t)k+t,k\bigr)=\Theta(n^t)$ for all $r>t\ge 2$ and $k\ge 2$. However, the existence and the value of the leading coefficient have remained largely open. We determine the coefficient $\pi(r,t,k):=\lim_{n\to\infty}n^{-t}f^{(r)}\bigl(n;(r-t)k+t,k\bigr)$ for every such $r,t,k$, except when $(r,t)=(3,2)$ and $k\ge 4$ is even. In particular, \[ \pi(r,t,k) = \begin{cases} \frac{2}{t!\bigl(2\binom{r}{t}-1\bigr)}, & \text{if $k$ is odd},\\ \frac{1}{t!\binom{r}{t}}, & \text{if $k$ is even and $r\ge4$}. \end{cases} \] Surprisingly, for $r\ge4$, the limit $\pi(r,t,k)$ depends on $k$ only through its parity, not its value. For the remaining case, we prove that $\pi(3,2,k)>1/6$ for all even $k\ge 4$, showing that the natural packing construction is never asymptotically optimal. As an application, we extend a connection of Bennett, Cushman, and Dudek to arbitrary uniformity to resolve several cases of the Erdős–Gyárfás–Shelah generalized Ramsey problem.

We study local systems on line arrangements for which no admissibility conditions are satisfied. In doing so we demonstrate the existence of these anti-admissible objects in both the finite and infinite setting and discuss their relation to the combinatorial data of arrangements. We also observe the connection between anti-admissibility and smooth branched coverings and how this connection is related to the branch set being minimal in one sense and maximal in another.

We give a deterministic algorithm that solves a nonsingular linear system $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ has $m$ nonzero entries and condition number $\kappa$, to any relative residual tolerance $0<\epsilon\le1/2$ using $\widetilde{O(}mn\log(\kappa/\epsilon))$ bit operations for inputs with logarithmically many bits per entry. For sparse, polynomially conditioned systems with $m=\widetilde{O}(n)$, this gives an $\widetilde{O}(n^2)$ algorithm for any inverse-polynomial accuracy, improving on the algorithm of Peng and Vempala, as sharpened by Nie, whose running time in this regime is approximately $O(n^{2.2707})$ with the best current matrix multiplication exponent, and largely closing a gap between the idealized performance of the conjugate gradient method in exact arithmetic and the running time achievable with finite-precision computation that has persisted for over 70 years. The algorithm is surprisingly simple. For integer inputs, we apply Dixon's lifting algorithm to the perturbed system $(A+I/R)x=b$ for a suitable integer $R$. After scaling, the matrix of this system is $RA+I\equiv I\pmod R$, so its modular inverse is trivial, and each lifting step needs only a sparse matrix-vector product with $A$ on $O(\log R)$-bit numbers. Fast rational reconstruction then recovers the exact solution of the perturbed system, which is an $\epsilon$-accurate solution of the original one. Normalization and rounding extend the result to fixed-point and floating-point inputs, with floating-point outputs represented using short integer significands and a common encoded exponent. A computable certificate removes the need for prior knowledge of $\kappa$.

We introduce and numerically demonstrate receiver-informed transmitter (RITX) waveform design for 200 Gb/s PAM-4 C-band direct-detection PON. Using 50 GHz optoelectronics, RITX achieves N1/E1 loss classes with FFE-Free ONU, enabling 40 km reach and 48.1%/66.7% network-wide complexity savings at 1:64/1:128 splits

We develop a rich theory of dyadic-probabilistic multiresolution methods in the context of flag type singular integrals. We apply these methods to prove a full $T1$ theorem, even a representation theorem, for bi-parameter flag singular integrals. This is the first time a complete $T1$ theorem appears in a multi-parameter setting with an entangled nature; even the related recent advances with Zygmund dilations have been in the cancellative $T1 = 0$ or convolution situations. This theorem comes with a non-trivial necessity argument related to the flag type product $\mathrm{BMO}$ assumption on $T1$. We also present a version of our theorem in the cancellative tri-parameter flag setting showing that our methods are perfectly adaptable to general multi-parameter flag settings. Importantly, our $T1$ representation theorem implies weighted estimates for every flag $A_p$ weight, showing that our dyadic multiresolutions carefully retain the flag invariance of the original singular integrals. Our multiresolution methods also yield flag style paraproduct decompositions of products of functions and lead to a general and efficient method for proving commutator estimates in various flag settings.

Ship structural vibrations contribute to noise, fatigue, and equipment damage, while dynamic-compliance topology optimization can produce pathological designs near resonance. This study extends neural-reparameterized topology optimization using a convolutional Kolmogorov-Arnold network (KATO) to forced-vibration design with active input power (AIP) as the objective. Applications include a 100 Hz engine-supporting deck panel and an 18 Hz thruster foundation frame. Helmholtz PDE filtering and Heaviside projection control feature sizes and manufacturing tolerance. Across both deck families, all eight optimized layouts reduce AIP relative to size-optimized references and, after finite-depth extrusion, also achieve lower static compliance. For unrestricted, manufacturing-aware, and stress-aware frame variants, KATO matches GCMMA in AIP within 0.5 dB while yielding 22-36x lower static compliance after matched-volume binary re-analysis. In a near-resonant 300 Hz case, both methods reduce initial AIP by more than 32 dB; KATO maintains a connected design, achieves 59x lower binary static compliance, and reduces maximum AIP over 1-500 Hz by 2.7 dB. KATO runs 6.4-10.4x faster than GCMMA for the implemented stress-aware formulations. The results demonstrate neural AIP-driven topology optimization as an efficient approach for designing connected, feature-size-controlled ship structures with improved forced-vibration performance.

Unit interval graphs admit a classical Catalan encoding by area sequences, or equivalently by Dyck paths. We introduce a new statistic on these graphs, called peak-nesting, defined as the largest number of peak-cliques containing a common vertex. Through the correspondence with area sequences, peak-nesting also defines a new statistic on Dyck paths. We construct a bijection between Dyck paths and bicolored Motzkin paths which simultaneously records peak-nesting and the number of peaks. This yields a refinement of Touchard's identity, coefficient formulas involving Dyck paths of given height, and new combinatorial interpretations for several sequences recorded in the OEIS. We study the peak and peak-nesting polynomials over all unit interval graphs and over the subclasses of connected, Abelian, $2$-nested, and reduced unit interval graphs. For these families we obtain closed formulas, recurrences, and, for the symmetric cases, nonnegative expansions in the gamma-basis with explicit combinatorial interpretations. We also investigate questions regarding the location of zeros and the log-concavity of the corresponding coefficient sequences: some of the polynomial families form generalized Sturm sequences, whereas others fail to be real-rooted but appear nevertheless to have log-concave coefficients. Finally, we relate the peak-clique presentation of a unit interval graph to lattice path matroids.

We show that the optimal commuting operator correlation for a variant of Coladangelo's two-player generalized nonlocal game, based on the embezzlement of entanglement, is a commuting operator self-test. Our result gives the first correlation self-test for an exact entanglement embezzlement protocol in the commuting operator framework. Because the unique optimal correlation necessitates the exact embezzlement of entanglement, the correlation is unrealizable using quantum tensor-product models, even if the local spaces are allowed to be infinite dimensional. As such, our result provides an example of a commuting operator self-test where the Hilbert space shared by the two parties cannot be factored into a tensor-product of local spaces that is compatible with the local measurement operators. Our main proof technique relies on Harris' unique abstract state characterization of exact embezzlement of entanglement involving Cuntz algebras.

We study the deterministic query complexity of finding approximate Goldstein stationary points of globally Lipschitz functions that may be nonsmooth and nonconvex. Under prescribed bounds on the Lipschitz constant and on the difference between the initial function value and the infimum, we prove a lower bound on the worst-case number of oracle calls that is exponential in the dimension at fixed sufficiently small accuracy parameters. The result holds for a local oracle and gives explicit dependence on the accuracy parameters. A complementary deterministic algorithm using only function values gives an exponential upper bound, establishing the exponential order in the dimension in this regime. For a coarser stationarity requirement, we also give a deterministic first-order algorithm with a dimension-free query bound. Thus, the dimension dependence differs between the two accuracy regimes.

Given integers $k,\ell \geq 2$, define the Erdős–Selfridge curve \begin{align*} \mathcal{C}_{k,\ell} : y^\ell = x(x+1)\cdots (x+k-1). \end{align*} Bennett and Siksek proved that if $(x,y)$ is a rational point on $\mathcal{C}_{k,\ell}$ with $\ell$ a prime and $y \neq 0$, then $\ell$ is bounded (doubly-exponentially) in terms of $k$. We improve this bound on $\ell$ when $k$ is sufficiently large. Our method uses combinatorial and sieve-theoretic arguments, as well as results on solutions to generalized Fermat equations.

We review mathematical results about the spectrum of spin-$s$ chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin-$s$ degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter $d=2s+1$. We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin-$\tfrac12$ $XXZ$ chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is $1-2/d$. Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers $[L+1]_q$ for $q+q^{-1}=d$. We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.

We propose a continuous-time algorithm for zeroth-order global optimization of smooth functions on compact manifolds that synthesizes Riemannian gradient dynamics using only objective measurements, without evaluating gradients. The algorithm is formulated as an ordinary differential equation driven by piecewise-smooth McShane-type approximations of the Wiener process, with a geometric compensation term that depends only on the actuation frame and not on the objective function. Our main result establishes infinite-horizon practical probabilistic guarantees for fixed approximation parameters. To this end, we show that the algorithm's trajectories approximate, in the mean-square sense, a Riemannian Langevin diffusion whose unique invariant Gibbs measure concentrates on the set of global minimizers. We exploit this approximation to establish finite-horizon probabilistic convergence guarantees and directly analyze the algorithm's long-time behavior to obtain the infinite-horizon result.

We classify the deep points of cluster varieties of cluster algebras of punctured marked surfaces with boundary.

We are interested in the molecule reduction algorithm used by Deng, Hani and Ma. This algorithm allows them to establish a rigidity theorem, which plays a central role in the long-time derivation of the Boltzmann equation using the hard-sphere dynamics. In the present article, we show that this algorithm is a graph traversal algorithm of Kruskal type, and we prove that it constructs a Kruskal spanning tree of the input molecule.

We study generalized learning dynamics in multi-agent systems whose joint state evolves on a manifold and whose agents act through state-dependent, potentially nonholonomic vector fields. Under a bundle-splitting condition, we show that these dynamics admit an intrinsic representation as projected Riemannian gradients, giving rise to a class of nonholonomic gradient play dynamics. We characterize the local stability of its equilibria through an intrinsic linearization that explicitly captures the effects of the Riemannian connection and the nonholonomy of the actuation frame. The framework recovers classical gradient play on Euclidean spaces and manifolds as special cases while accommodating nonholonomic actuation. We then develop a geometric theory of deception under asymmetric information, whereby an agent exploits privileged knowledge of other agents' learning rules to manipulate the emerging equilibrium. We show that deception effectively tilts the Riemannian geometry perceived by the oblivious agents, distorting their projected gradient directions. We establish persistence of exponentially stable equilibria over an open set of deception parameters and derive an explicit first-order characterization of the resulting equilibrium displacement. Analytical and numerical examples illustrate the framework and its implications for equilibrium manipulation in games.

Beke, Csáji, Csikvári, and Pituk showed that the Merino–Welsh quotient $\Phi(M)=T_M(1,1)^2/(T_M(2,0)T_M(0,2))$ can be arbitrarily large, so the multiplicative Merino-Welsh inequality fails for matroids in general. We show that $\Phi$ is uniformly bounded on the class of matroids whose cyclic-flat lattice avoids any fixed finite poset $P$ as an induced subposet. We further prove $\Phi(M)\leq1$ for matroids of cyclic width at most $7$, cyclic height at most $6$, and for loop- and coloop-free $4$-paving or $4$-copaving matroids.

We reveal an unexpected connection between the parallel Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. The twirling channel induced by the parallel Kac's walk on the symmetric subspace is exactly encoded by a classical Markov chain on partitions, which lifts to a parallel KMP process on complete graphs. This correspondence reduces the analysis of the twirling channel to the mixing of the parallel KMP process. We prove that $O(\log d+\log(1/\varepsilon))$ repetitions suffice to approximate Haar twirling on the symmetric subspace of $(\mathbb C^d)^{\otimes t}$ to error $\varepsilon$, uniformly in the number of copies $t$. For the standard KMP process on general graphs, we prove a mixing-time analogue of Aldous's conjecture: at fixed accuracy, the mixing time of the $t$-particle process is at most a constant times the single-particle mixing time multiplied by the logarithm of the number of vertices, uniformly in $t$. As an application, we improve the total variation mixing-time bound for coordinate hit-and-run on the $n$-dimensional standard simplex from $\widetilde O(n^3)$ (Kook and Vempala, 2026) to $\widetilde O(n)$, while removing the dependence on the initial distribution. Our main technical contribution is conditional product structure for both parallel and standard KMP processes. Conditioned on suitable auxiliary randomness, the labeled particles evolve independently. Combining this structure with an exact coupling yields mixing bounds uniform in the number of particles for both unlabeled KMP models. These bounds are sharp up to logarithmic factors and imply rapid convergence of the parallel Kac twirling channel on the symmetric subspace.

We prove a sharp Santaló-type inequality for the product of $V(\varphi)$ and $V$ of the Legendre transform of $\varphi$, where $\varphi$ is an even geometric convex function on $\mathbb{R}^n$ and $V(\varphi)=\int(1+\varphi)^{-(n+1)}$. For this size functional, the Legendre transform and functional polarity give the same volume, so the same inequality holds for both dualities. Through a projective correspondence, the problem is equivalent to a new Santaló-type inequality with respect to standard volume and standard polarity, on a class of not necessarily centered convex bodies. This class consists of convex bodies $K\subset\mathbb R^n\times[-1,1]$ containing $\{0\}\times[-1,1]$, for which every horizontal section is centrally symmetric about the distinguished vertical axis. We determine the sharp upper bound for $\operatorname{vol}_{n+1}(K)\operatorname{vol}_{n+1}(K^\circ)$ in this class and classify all equality cases. When $n=1$, the extremizers are the disk and its horizontal linear images. When $n\ge 2$, the ball is no longer a maximizer, and the extremizer is unique up to horizontal linear transformations and the reflection $t\mapsto -t$. The symmetric Santaló inequality applied to horizontal sections, together with monotone transport, reduces the problem to a maximization problem for increasing curves in the square $[-1,1]^2$. We solve this problem using a global potential when $n=1$ and two Hamilton-Jacobi branches when $n\ge 2$. Finally, in dimension one we provide a sharp family of functional Santaló inequalities for the Legendre duality, interpolating between the volume $V$ and the exponential volume $\int \exp(-\varphi)$, with centered quadratics as the only equality cases.

For real codimension two smooth foliations, we study the transversely complex structures, transversely complex projective structures and transverse opers associated to the foliation. We prove a uniqueness theorem for the transversely holomorphic ${\mathbb C}{\mathbb P}^1$–bundle naturally associated with a transversely complex projective structures. A similar uniqueness theorem is proved for the transversely holomorphic filtered ${\mathbb C}{\mathbb P}^{r-1}$–bundle naturally associated to a transverse ${\rm PGL}(r,{\mathbb C})$–oper.

We give an explicit solution to the five expert prediction with expert advice partial differential equation (PDE) in the finite-time horizon setting. The solution formula establishes that the adversary's rank strategy $(1,0,1,0,0)$ is globally optimal, and the COMB strategy $(1,0,1,0,1)$ is optimal exactly on the set where $x_1=x_2$ and $x_3=x_4$. The formula is derived from the solution of the geometric-stopping problem given in our companion paper through the transform principle of Bayraktar, Ekren and Zhang, which links the two problems by a Laplace transform. Inverting the transform term by term expresses the solution through a series of Gaussian and complementary error function kernels. The optimality of $(1,0,1,0,0)$ is reduced to the signs of $41$ one-variable Gaussian series, which are certified with computer assistance by Poisson summation, first-mode domination and interval arithmetic on $1616$ rational cells. The proofs of our main theorems, certificates included, are also formalized in the Lean proof assistant.

We prove a conjecture proposed by the first-named author in 1998. Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function $N_1(r,f)$ of its Wronskian zeros satisfies $N_1(r,f)=o(T(r,f))$, then its order and lower order coincide and belong to $\{1+k/q:k\in\mathbb{Z}_{\geq 0},\ 2\le q\le n+1\}$, and its characteristic is regularly varying. Every order in this set occurs. We also prove the sharp inequality $\limsup_{r\to\infty}N_1(r,f)/T(r,f)\geq 1$ for transcendental linearly non-degenerate curves of order zero.

We prove weak solvability for a two dimensional nonlinear Boussinesq type groundwaterflow problem motivated by the Moche CHAVIMOCHIC aquifer in northern Peru. The model couples state dependent storage and transmissivity with a boundary decomposition into Dirichlet, Neumann, and nonlinear Robin parts dictated by the hydrogeological geometry. In contrast with much of the groundwater Boussinesq literature, which is devoted to exact, similarity, perturbative, semi-analytical, or linearized solutions for special geometries and boundary data, our objective is a variational existence result for a genuinely two-dimensional mixed-boundary problem. Because the physical transmissivity is proportional to the saturated thickness, the equation degenerates at local depletion. We therefore isolate a physically admissible saturated range and construct uniformly positive extensions of the constitutive laws. For the resulting uniformly parabolic problem, existence of a weak solution is obtained through an implicit Euler Rothe scheme, a nonlinear energy adapted to the storage law, uniform estimates for the discrete derivative of the storage potential, discrete compactness, and strong convergence of boundary traces. We then give a precise consistency statement showing when the abstract solution solves the original Moche model and identify the fully depleted regime as a distinct porous medium type problem.

We investigate the recently introduced polygonal hypergeometric functions [arXiv:2502.12127, arXiv:2507.01904], which are conjectured to evaluate multipoint parametric one-loop conformal integrals in arbitrary dimensions. These functions can be systematically constructed using a diagrammatic algorithm that operates in terms of simple planar shapes. Given an $n$-point polygonal power series, we formulate the corresponding GKZ hypergeometric system, which admits an explicit diagrammatic interpretation. In particular, the transianic matrix, which encodes all the relevant diagrammatic data, determines the corresponding toric matrix along with a basis of the lattice of relations, thereby singling out a finite subsystem of $n(n-3)/2$ toric equations. This finite subsystem consists of second- and third-order equations, whereas the full toric family also includes higher-order ones. We illustrate the general procedure with several prominent examples of polygonal functions, including the fourth Appell and the Srivastava–Daoust functions.

On-policy distillation (OPD) trains a student on its own generated responses using dense, token-level supervision from a stronger teacher. Vanilla OPD treats all teacher signals equally, assuming that the teacher's supervision is equally important for every token. However, teacher signals at different tokens may have very different effects on the student's performance: some correct important reasoning errors, while others have little effect on the final answer. Motivated by this observation, we introduce Dr. OPD (OPD Done Right), which defines the optimal weighted OPD to maximize the student's performance. We formulate Dr. OPD as a bilevel optimization problem in which the student learns from weighted teacher supervision, while the weights are selected to maximize the expected reward of the resulting student. To solve Dr. OPD, we develop an efficient iterative solver that updates the token weights and student policy alternatively. At each round, it updates weights in closed form and then takes one gradient step on the resulting weighted OPD objective. Under regularity conditions, we show that this weighted update achieves a higher expected reward than a vanilla OPD update. Empirically, across strong-to-weak and same-size distillation on math and code, Dr. OPD consistently outperforms all evaluated baselines. In particular, in the strong-to-weak distillation setting, Dr. OPD improves average math performance by $9.7$ points over vanilla OPD, and enables the smaller student to surpass its larger teacher.

For Bernoulli bond percolation on every regular tree of degree at least three, we prove that infinite clusters appear strictly before absolutely continuous spectrum. Throughout an explicit supercritical interval, the adjacency operator of every open cluster almost surely has no absolutely continuous component. Together with Bordenave's theorem, this gives $p_c<p_{ac}<1$.

We propose a single-level reformulation (SLR) for the pessimistic bilevel optimization problem that does not rely on complementarity conditions or optimal value functions. For this reason, we refer to it as a true single-level reformulation (tSLR). A remarkable consequence is that this reformulation can satisfy the classical linear independence constraint qualification, despite the fact that even the weaker Mangasarian-Fromovitz constraint qualification is known to systematically fail for standard single-level reformulations of both optimistic and pessimistic bilevel programs. We leverage on these constraint qualifications to construct new necessary optimality conditions for pessimistic bilevel optimization. The reformulation also has a striking limitation: under the assumptions of our analysis, the classical second-order sufficient condition fails at every Karush-Kuhn-Tucker point of the problem. Nevertheless, preliminary numerical experiments demonstrate that algorithms based on the proposed tSLR can outperform existing approaches for pessimistic bilevel optimization. Overall, the proposed framework suggests that pessimistic bilevel programs may be considerably more tractable than previously believed and need not be inherently more difficult to solve than their optimistic counterparts.

We construct an automorphism of a smooth projective K3 surface with positive topological entropy and a non-empty Fatou set. The Fatou set contains invariant domains biholomorphic to products of two annuli. On each domain, the automorphism is holomorphically conjugate to a fixed-point-free irrational rotation. These domains are two-dimensional analogues of Herman rings. The construction answers a question posed by McMullen in 2002 and several questions posed by Cantat in his 2018 ICM address.

We establish improved locally Markovian properties of quantum Gibbs states at every fixed positive temperature. For finite-range interactions of bounded interaction degree, the conditional mutual information (CMI) decays exponentially with separation, with a prefactor exponential in the boundary size of only one of the two regions and independent of the size of the other region and the total system. This improves the exponential volume dependence of the local Markov bounds established by Chen and Rouzé \cite{Chen2025}. For exponentially and stretched-exponentially decaying interactions, we obtain stretched-exponential CMI decay with a boundary-dependent prefactor. For power-law interactions, we prove inverse-logarithmic or algebraic decay with polynomial region-size dependence. Our proof is static, in contrast to the dynamical approach of Chen and Rouzé. We compare the Gibbs state with a product state obtained by removing boundary interactions, reducing the CMI bound to a relative-entropy loss that we control using locality estimates.

Lazard asked which $N$-series of a group arise from multiplicative filtrations of the integral group ring. Given an $N$-series $H_\bullet$ of a group $G$, let $A_\bullet$ be the filtration of the augmentation ideal spanned by products of elements $x-1$, $x\in H_a$, weighted by $a$. Every realizing filtration contains $A_\bullet$, so $H_\bullet$ is realizable exactly when the maps $H_n/H_{n+1}\to A_n/A_{n+1}$, $h\mapsto h-1$, are injective. We show that each kernel is the cokernel of an explicit map of normalized bar groups; for a finite group this makes realizability decidable by integer linear algebra. For a profinite group with a cofinal tower of finite quotients and an $N$-series induced from it, suppose $h\in H_n$ and $h-1\in A_{n+1}$ in every finite quotient. Then $h-1\in A_{n+1}$ in the discrete group ring exactly when the widths of its finite-level Losey expressions are bounded. This descent can fail. Using Tahara's class-three $2$-groups and the Hartl-Mikhailov-Passi description of the fourth dimension subgroup, we construct a countable product $P$ of finite $2$-groups. It contains an element of $\gamma_3(P)$ that lies in the fourth dimension subgroup of every finite subproduct of $P$ but not in $D_4(P)$.

We prove a geometric version of Poonen's "Mordell–Lang plus Bogomolov" theorem for semi-abelian varieties in characteristic zero. This is a generalization of the Mordell–Lang conjecture and the geometric Bogomolov conjecture.