Papers

Let $K>1$ be a real number and let $A\subsetneq \mathbb Z$ be a finite and nonempty subset. Suppose $|A+A|<K|A|$. A fundamental open problem is to characterize the size of higher iterated sumsets $nA = A+\cdots+A$ ($n$-fold sumset). In 2004, Terry Tao formally posed the problem of determining the optimal growth function governing iterated sumsets of small-doubling sets. In this manuscript, we give a solution to this long-standing open problem for large enough $n$.

We give a negative answer to the second question of Erdős Problem #1117: can the number $\nu_f(r)$ of maximum-modulus points of a nonzero, non-monomial entire function on $|z|=r$ tend to infinity as $r\to\infty$? Write $f(z)=z^m g(z)$ with $g(0)\ne0$, and let $k$ be the first positive exponent with a nonzero coefficient in the Taylor expansion of $g$. We prove that $\{r>0:\nu_f(r)>2k\}$ is locally finite in $(0,\infty)$, and that the constant $2k$ is sharp. In particular, $\liminf_{r\to\infty}\nu_f(r)\le2k<\infty$, without a growth assumption. The proof combines finite local parametrizations of a logarithmic-derivative correspondence with analytic continuation and a local description of stationary points.

Fix $0<\kappa<8$ and put $d=1+\kappa/8$. Recent work by OpenAI identifies an explicit exact Hausdorff gauge \( h_\kappa(r)=r^d\bigl(\log\log(1/r)\bigr)^{(2-d)/2} \) for sufficiently small $r$ for chordal $\mathrm{SLE}_\kappa$: the corresponding Hausdorff measure is almost surely positive and finite on every segment $\gamma([s,t])$ with $0<s<t<\infty$, and has finite expectation in bounded subsets of the plane. We show that this Hausdorff measure is a deterministic positive constant multiple of the occupation measure (natural length) constructed by Lawler and Rezaei.

The polynomial threshold degree (or threshold degree) of a Boolean function is the least degree of a real polynomial whose sign represents it. We prove that almost every Boolean function has polynomial threshold degree among the central one or two integers. More precisely, a uniformly chosen Boolean function on $n$ variables has threshold degree $\lfloor n/2\rfloor$ or $\lceil n/2\rceil$ except with probability at most $\exp(-c2^{n/2}/n)$, for an absolute constant $c>0$ and all sufficiently large $n$. A parity example gives a lower bound on the exceptional probability whose exponent differs from ours by a factor of order $n$. The proof relates polynomial sign representation to recovery of Reed–Muller codes from random erasures, and combines sharp-threshold estimates with bounds on generalized Hamming weights and minimum distance.

In this paper, we prove two conjectures concerning parametric $q$- supercongruences related to Van Hamme's $\mathrm{(G.2)}$ supercongruence. First, we confirm Tang\rq s conjecture \cite[Conjecture $4.1$]{Tang G2} by establishing the corresponding congruences modulo $[m]\Phi_m(q)^2$. Second, we prove a conjecture of Wang and Xu \cite[Conjecture $2$]{WangXu24}, extending their $q$-supercongruence modulo $\Phi_m(q)^5$ to modulo $[m]\Phi_m(q)^4$ when $m\equiv1\pmod4$.

Variational regularization techniques are dominant in the field of mathematical imaging. The expressiveness of these methods heavily depends on the adopted regularization formulation which usually involves a number of parameters which have to be fine-tuned by the user. To circumvent the heavy computational costs associated with this parameter tuning, data-driven learning strategies have turned out to be a useful strategy. Selecting parameters by learning from data gives rise to a hierarchical (bilevel) optimization problem, which is in general computationally hard to solve. Our aim in this paper is to develop a data-efficient solution strategy for the bilevel learning problem. To that end, we propose a multilevel Monte Carlo derivative-free optimization method that couples Gaussian smoothing of the upper-level objective (parameter selection) with a hierarchy of lower-level (variational regularization) solution accuracies. Being agnostic towards the specific way the lower-level problem is solved, we derive explicit oracle complexity and computational cost statements for approximately solving the entire bilevel learning problem. Our statements are parameterized by the employed lower-level solution method, thus allowing for a direct cost accounting from the variational reconstruction method to the parameter selection problem. This enables us to give a universal end-to-end complexity analysis of bilevel learning, valid for the most important optimization routines in inverse problems, rather than for a specific choice of methods.

We study the recovery of a spatial coefficient in the principal memory flux of a three-dimensional wave equation with the prescribed exponential relaxation kernel $\beta e^{-\beta t}$, where $\beta>0$ is known. From a single finite-time lateral Cauchy measurement generated by one fixed, coefficient-independent initial displacement that is affine near the unknown region, with zero initial velocity, we establish global uniqueness and conditional $L^2$-Hölder stability. Two difficulties arise in this setting. The affine initial displacement makes the initial acceleration vanish in the unknown region, while the coefficient-difference equation contains a divergence-form source whose direct $L^2$ estimate loses a spatial derivative. We establish an acoustic Carleman estimate that controls the vector source and its time derivative without spatially differentiating the source. The affine condition identifies the relevant initial trace with a directional derivative of the coefficient difference, and the quadratic temporal vanishing of the source permits absorption in the weighted estimate. A transport estimate then yields stability uniformly over a bounded, uniformly coercive $W^{5,\infty}$ coefficient class, without an ordering condition or a smallness assumption near a reference coefficient.

We prove that for every integer $n>2$ the largest good lattice point matrix $U_n=[u_h:h\in H_n]$, whose columns are generated by all integers $h<n$ coprime to $n$, has rank exactly $\varphi(n)/2+1$. The upper bound follows from the pairing identity $u_h+u_{n-h}=(n,\dots,n,2n)^{\top}$. For the lower bound the columns are centered and the resulting orbit under the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is analysed by means of odd Dirichlet characters; the essential arithmetic input is the non-vanishing of $\sum_{a}a\,\psi(a)$ for odd primitive characters $\psi$, which is equivalent to Dirichlet's theorem $L(1,\psi)\neq 0$.

The rectangular multiparameter eigenvalue problem (rMEP) is a recently introduced form of an eigenvalue problem, in which we are looking for the combinations of parameters where the multivariate rectangular matrix loses full column rank. Existing numerical methods for rMEPs typically require substantial memory and are not suitable for problems with a large number of eigenvalues. We propose a novel numerical approach based on the homotopy method that has low memory requirements and can compute all eigenvalues of an rMEP. Moreover, the homotopy method is naturally parallelizable, providing a simple and efficient way to accelerate computations. An implementation, together with a collection of test examples, is available in a public repository.

Real-world Vision-Language Navigation (VLN) requires agents to act under partial observability, where map structure, semantic cues, and target locations may be incomplete or incorrectly inferred by foundation models. This paper presents a lightweight, training-free, and uncertainty-aware framework for cooperative multi-agent multi-target search in mapless environments. Instead of learning an end-to-end navigation policy, the framework separates high-level semantic reasoning from low-level execution through a scalable graph-optimization backbone. Multimodal Large Language Models (MLLMs) serve only as hypothesis generators, proposing semantic zones, target-location cues, and possible navigable connections. These outputs are treated as uncertain priors and further refined using online observations and graph-context evidence. Cooperative navigation is then formulated as a rolling-horizon multi-agent mixed-integer linear program with a lexicographic objective that first maximizes target-directed reward and then reduces travel distance, uncertain-edge use, and repeated visits. Experiments derived from 100 Matterport3D test cases show that the proposed framework improves target coverage and search efficiency over representative baseline methods, while shared information and joint planning enable more balanced workloads across agents. By operating on an abstract graph without relying on embodiment-specific policy parameters, the framework scales across varying numbers of agents and targets, generalizes to unseen environments, and accommodates different perception modules and low-level controllers without task-specific retraining.

Let $p$ be a prime and $k$ be an algebraic closure of the finite field with $p$ elements. Let $G$ be a connected reductive algebraic group over $k$, and let $F:G\rightarrow G$ be a Frobenius morphism, defining an $\mathbb{F}_q$-rational structure on $G$, where $q$ is a power of $p$. Let $\ell$ be a prime different from $p$ and $\ell\ne 2$. Let $\Pi$ be an $\ell$-adic cuspidal unipotent representation of $G^{F^\ell}$ such that $\Pi\simeq \Pi^\sigma$, where $\sigma$ is a generator the Galois group ${\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$. In this paper, we prove the irreducibility of the zeroth Tate cohomology group $\widehat{\rm H}^0(\sigma,\mathcal{L})$ and vanishing of the first Tate cohomology group $\widehat{\rm H}^1(\sigma,\mathcal{L})$ for any $G^{F^\ell}\rtimes{\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$-stable lattice $\mathcal{L}$ in $\Pi$, using the multiplicity-one property of a progenerator constructed via the generalized Gelfand-Graev representations and Harish-Chandra induction.

This dissertation has two parts. In the first part we theoretically develop a framework that extends the optimal transport barycenter problem from independent data with observed covariates to processes whose relevant covariate is their own history. Given a stationary stochastic process, our framework shows there exists an invertible transport map that extracts from it an iid component whose distribution is the barycenter of the conditional distributions of the input process on its own past. This extracted iid sequence thus captures the innovation or noise component of the process that is not explained by the process's own past. In the second part of this thesis we implement an effective algorithm to solve this problem and test it against synthetic data generated with some well-known models in time series analysis for which the solution is known in closed-form. We then show how this method can be applied to build a predictive sampler for such processes, and assess the accuracy of this predictive sampler both with the aforementioned synthetic data and with real-world time series such as the S&P 500 and Bitcoin 4-hour returns, and the daily records of the North Atlantic Oscillation index from the NOAA Physical Sciences Laboratory. Finally, we demonstrate an application of the sampler to path simulation with a well-known time series model.

We study Markowitz's mean-variance portfolio optimization problem. When practically using this model, the mean vector and the covariance matrix of the assets returns often need to be estimated from the sample data. The sample errors will be propagated to the optimization output. In this paper, we consider three commonly used mean-variance models and build the asymptotic properties for the conventional sample approximations that are widely adopted and studied, by leveraging the stochastic optimization theory. We show that for all three models, under certain conditions the sample approximations have the desired consistency and achieve a convergence rate of square root of sample size, and the asymptotic variance depends on the first four moments of the returns. To address the variability and the uncertainty in the data driven portfolio optimization, we further consider the distributionally robust optimization (DRO) for the mean-variance model. For the ambiguity sets constructed by the $\phi$-divergence and the Wasserstein distance, we build the tractable reformulations for the DRO models. We conduct numerical experiments to test the asymptotic properties for the estimation. We also conduct experiments to illustrate that the asymptotic normality might not hold when the fourth moments of the returns do not exist.

A bisection of a graph is a partition of its vertex set into two parts whose sizes differ by at most one. We prove that every graph with $m$ edges and maximum degree at most $k$ has a bisection of size at least $\frac{k+1}{2k}m$ when $k$ is odd, and at least $\frac{k+2}{2(k+1)}m$ when $k$ is even. Both bounds are sharp, with equality for complete graphs. This establishes the unweighted case of the bisection conjecture of Gerke, Gutin, Yeo and Zhou (arXiv 2024), removes the additive error from the corresponding bounds of Lee, Loh and Sudakov (JCTB 2013), and extends the odd-degree regular case proved by Bollobás and Scott (JGT 2004) to arbitrary graphs of bounded maximum degree. The proof combines the matching-and-pairing method with the Gallai–Edmonds decomposition. Hall's theorem controls the number of uncovered factor-critical components whose internal degrees are all even. A boundary-edge count bounds this obstruction by the total degree deficiency, while the remaining uncovered components supply the required pairing gain.

In this paper, sufficient conditions are established for diagonal perturbations of weighted shifts to be supercyclic, hypercyclic, and frequently hypercyclic, extending several classical results for weighted shifts to this more general setting. In addition, a complete analytic characterization of the shadowing property for these operators is obtained. Consequently, examples of frequently hypercyclic perturbed weighted shifts with the shadowing property are constructed.

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $\phi=b/a$, where $a$ is the Pythagorean mate of $b$. In an earlier article, the author showed that, if $\phi\in H^2$, then \[ f\in\mathcal{H}(b) \iff \Bigl(f\in H^2 \text{ and }\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr), \] in which case we have \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2. \] It was also shown that there exists a space $\mathcal{H}(b)$ for which $\phi\in\cap_{p<2}H^p$, yet \[ f\in\mathcal{H}(b)\not\Rightarrow \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty. \] Here we complete this circle of ideas by showing that there exists a space $\mathcal{H}(b)$ for which $\phi\in\cap_{p<1}H^p$, yet \[ \Bigl(f\in H^2\text{ and } \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr) \not\Rightarrow f\in\mathcal{H}(b). \]

A central object in the theory of proof complexity generators is Krajicek's local witness model. We construct a new such model as a definable ultrapower using a definable version of a selective ultrafilter over a model of arithmetic. We transfer some set theory of combinatorial ultrafilters to the definable nonstandard setting.

We show that uniqueness may fail for one-dimensional McKean–Vlasov stochastic differential equations with Hölder continuous diffusion coefficients. More precisely, for every \(\alpha\in(1/2,1)\), we construct an equation with a diffusion coefficient that is \(\alpha\)-Hölder continuous in the state variable, Lipschitz continuous in the measure variable, and of linear growth, for which uniqueness in law and pathwise uniqueness fail. The construction relies on a spreading estimate together with a Schauder fixed-point argument.

Rural last-mile logistics remains challenging because of sparse demand, limited road infrastructure, and long travel distances. We study a Truck-as-Hub Vehicle Routing Problem with Drones in Rural Areas, in which multiple trucks each carry a dedicated fleet of drones. Drone launch and retrieval are restricted to predesignated passive synchronization pads, where a visiting truck temporarily acts as a mobile hub, may dispatch multiple drones simultaneously, and must wait until all launched drones return before continuing its route. We formulate a mixed-integer quadratic programming model that jointly accounts for customer delivery delays, truck return times, and operating costs. To solve the problem exactly, we transform the original truck–drone network so that each feasible route can be represented as a single path and develop a branch-price-and-cut algorithm with tailored pricing, cutting, branching, and parallel-computing procedures. Computational experiments show that the proposed algorithm proves optimality for most large-scale test cases and achieves optimality gaps of no more than 4% for the remaining cases, whereas Gurobi reaches the 1,800-second time limit on all large-scale instances when applied directly to the compact formulation. Sensitivity analyses show that drone operating costs and managerial priorities substantially affect drone utilization and customer delays, providing practical guidance for balancing service quality and operating cost in rural delivery systems.

We prove that, in every dimension, a bounded, non-constant entire solution of the Allen–Cahn equation whose zero set is contained in a half-space is a one-dimensional heteroclinic solution.

In this article, we prove a stability result for the identification of the coefficients appearing in a coupled relativistic Schrödinger system posed on a bounded domain of $\mathbb{R}^{1+n}$, where $n \geq 3$. More precisely, by utilizing input-output operator, where Neumann boundary data are available only on a subset of the boundary, and constructing an appropriate geometric optics solution, we derive a stability estimate of $\log$-$\log$ type for the matrix-valued first-order coefficients. This result is subsequently used to obtain a $\log$-$\log$-$\log$ type stability estimate for the zeroth-order matrix-valued potential.

In this follow-up to \cite{Sua25}, we extend the results to the coupled anomaly flow which incorporates the evolution of a Hermitian metric on a holomorphic vector bundle over a compact complex $3$-fold with general slope parameter $\alpha'$. This yields a similar sufficient smallness condition on the slope parameter $\alpha'$ in order to extend the flow on $[0, \tau)$ to $[0, \tau + \epsilon)$. We also augment the argument of \cite{Sua25} slightly to fix the peculiarity of the units in the condition for $\alpha'$ for the $k = 1$ case.

We present DisCo algebra—a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a symmetric operad (combination) on the bare family $\mathcal{P}_n=\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not—the information loss is exactly the conditional KL divergence between the true conditional distribution and the conditionally independent product. This asymmetry reveals a hard limit on how much cross-block dependence can be captured by scalar block summaries.

We study the exponential Green energies associated with a finite point configuration on a Riemann surface. We focus on studying optimal or close-to-optimal configurations, and show that they must be well-spread out on the surface. We study in particular the round sphere and obtain values for the minimal Green energy of a configuration. The first version of this paper was written with the help of ChatGPT.

We prove a compact two-weight $T1$ theorem for the $\alpha$-fractional Riesz vector, $0\leq\alpha<n$, when one measure is supported on a compact regular $C^{1,\mathrm{Dini}}$ curve $\Gamma$. The measures may have common point masses. Under the global boundedness assumptions, we give a two-sided estimate for the essential norm in terms of Muckenhoupt and testing constants computed after restricting both measures to small cubes. Compactness is equivalent to the uniform vanishing of these local bounds as the cube side length tends to zero. The smooth truncations have a unique weak operator limit and converge in operator norm in the compact case. For arclength measure and distance power weights, we determine the compactness ranges for Green gradients on bounded tubes and for Newtonian gradients and Biot–Savart fields on the whole space. The whole-space operators in these ranges are compact but are not Hilbert–Schmidt. When $\alpha=n-1$ and both measures have positive continuous arclength densities $a,b$ on the same closed curve, we obtain the exact essential norm $\pi c_{n-1,n}\max_\Gamma\sqrt{ab}$.

Richardson varieties provide geometric representatives for products of Schubert classes, with stratifications indexed by the subintervals of Bruhat intervals. We construct analogous geometric models for tilted Bruhat intervals for all split connected reductive groups. Our construction uses Richardson varieties in affine flag varieties and yields abstract tilted Richardson varieties whose isomorphism classes, together with their labeled stratifications, are independent of the choices involved. We obtain an explicit formula for the point-counting polynomials of their open parts in terms of increasing minimal weight paths, extending Dyer's formula for Kazhdan–Lusztig $R$-polynomials. Over the real numbers, their totally nonnegative parts are regular CW complexes homeomorphic to closed balls, which gives a natural geometric realization of the posets of tilted Bruhat subintervals in all Lie types. Over the complex numbers, we prove that their fundamental classes push forward under a continuous loop evaluation map to the Poincaré duals of the minimal-degree coefficients of quantum Schubert products.

We extend the packing results of Etayo, Hedenmalm, and Ortega-Cerda for complex tori to arbitrary compact Riemann surfaces. The construction uses two different approaches, one based on complex geometry of line bundles, and the other on Laplace designs. Both approaches give the same basic existence of close-to-optimal point configurations on the given surface. This version was written with the help of ChatGPT.

The average-reward setting for reinforcement learning is challenging because its Bellman operator is known to contract in span only when the underlying Markov chain is mixing. Whereas existing work typically assumes a contraction property, e.g., through environment mixing, we consider the setting where we are given a reward-agnostic $m$-step mixing policy, and we control the contraction and exploration properties of the learned policy by regularizing toward the mixing policy. We provide finite-time analysis of $Q$-learning algorithms that can be instantiated with multiple choices of regularization, with guarantees measured with respect to the regularized fixed point. Our framework exposes novel regularizer-dependent objects that control contraction, exploration, and operator bias. In the one-step setting, our rates match the best known rates for $Q$-iteration and synchronous $Q$-learning, and for $m$-step mixing we derive new rates. We also study $Q$-planning variants, and show that planning acts as time-localized learning, where the planning horizon $H$ trades off the bias from the boundary value against the sample complexity. Finally, we present numerical studies validating the analysis: (i) knowledge of a strong mixing policy improves convergence and exploration, and its advantage increases with bottleneck severity and, (ii) our theory predicts the convergence and bias rankings of regularizers from the $\alpha$-divergence family.

Let $E$ be the exterior algebra of an $n$-dimensional vector space over a field, let $F$ be a graded free $E$-module with $r$ generators of degrees $d_1 \le \dots \le d_r$, and let $M$ range over the monomial submodules of Amata and Crupi. The Hilbert function of $F/M$ is a shifted sum of $f$-vectors of simplicial complexes; the companion paper determines the convex hull of such sums, and here we count them. For $r = 1$ the count is Linusson's (1999); we show that no new recursion is needed for general $r$. Using two statistics of an $f$-vector – its top nonvanishing degree and the initial degree of the corresponding ideal – we prove that the number of achievable Hilbert functions is a product of $r-1$ transfer matrices, one per generator-degree gap, with entries differences of Linusson's refined counts $E^p(n,k)$. The rank enters only as the length of a matrix product. The same holds for the proper variant. Consequences: a linear recurrence in $r$ for constant gaps, with exact exponential asymptotics $K \lambda^r$ when the gap is positive; polynomiality in $r$ when all gaps vanish; a threshold beyond which the count is a plain power; and polynomiality in the number of vertices.

We prove Kahane's conjecture: if a continuous unimodular function $F$ on the circle satisfies $\|F^n\|_{A(\mathbb{T})}=o(\log|n|)$ as $|n|\to\infty$, $n\in\mathbb{Z}$, then $F(x)=e^{i(mx+b)}$ for some $m\in\mathbb{Z}$ and $b\in\mathbb{R}$. The main step is a logarithmic lower bound for averages: for every nonconstant real-valued $\varphi\in A(\mathbb{T})$, $\frac{1}{T}\int_a^{a+T}\|e^{it\varphi}\|_{A(\mathbb{T})}dt \ge c_\varphi\log T \quad(a\in\mathbb{R},\ T\ge T_\varphi)$, where $c_\varphi>0$ and $T_\varphi>1$ are independent of $a$. The proof uses exact sampling on finite subgroups of the circle, uniformly bounded localized modulations, and the weighted non-harmonic Littlewood inequality of Jaming–Kellay–Saba. A suitable cutoff and a small linear perturbation of the phase ensure that a fixed positive proportion of the coefficients at the beginning of the frequency ordering have modulus one. Passing to integer powers and removing the degree of the circle map yield Kahane's conjecture without any additional smoothness assumptions. We also give equivalent phase, convolution, and operator formulations, and derive consequences for homomorphisms of weighted Wiener algebras and for higher-dimensional tori.