Training reinforcement learning (RL) policies from scratch is costly: it requires careful reward and environment design, extensive tuning, and substantial computation. Yet many control problems already have a functional but suboptimal policy available as a baseline. This paper proposes a method for embedding such a baseline into the RL training process, simultaneously improving training efficiency relative to from-scratch methods and producing a learning policy that outperforms the baseline. At each step, the method arbitrates between the baseline policy and a trainable learning policy, initially relying strongly on the baseline policy and then progressively transferring agency to the learning policy. By the end of training, the learning policy is a standalone neural network that operates without baseline policy support. The paper formalizes what it means for the baseline policy to be functional: under this policy, the agent reaches a goal set and remains there with high probability. The proposed arbitration mechanism is designed to exploit this property during training, yielding high goal-reaching rates right from the beginning of training. A theoretical analysis provides a formal interpretation of this behavior under stated assumptions and extends it to the final baseline-free regime, where explicit lower bounds are derived for the goal-reaching probability of the standalone learning policy. Empirical results on continuous-control benchmarks show that the proposed method achieves returns that match or exceed those of competitive approaches, while maintaining the highest goal-reaching rates throughout training among the compared methods -- including in the final stage, where the learning policy operates without any baseline support.

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem (UAT) for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

In this note, we use a simple argument to show the existence of large values of conjecturally sharp size for Dirichlet $L$-functions attached to primitive characters of fixed order at $σ\in (1/2, 1]$. More precisely, for every fixed integer $g\geq 2$ we prove the existence of a primitive character $χ$ of order $g$ and conductor $Q\asymp x$ such that $|L(1,χ)| \geq e^γ\left(\log\log x+\log\log\log x-\log(2\log g)+o(1)\right). $ We also show that for every fixed $1/2<σ<1$ there exists a primitive character $χ$ of order $g$ and conductor $Q\asymp x$ such that $\log |L(σ,χ)| \geq \left(C_g(σ)+o(1)\right) (\log x)^{1-σ}(\log\log x)^{-σ}, $ for some explicit positive constant $C_g(σ).$ Previously, such bounds were known only conditionally on the Generalized Riemann Hypothesis, and even then only in the special cases $g=2$ and $g=3$.

Standard diffusion models typically use a single time-homogeneous Gaussian terminal distribution as the reference law for generation. While this choice is analytically convenient and empirically powerful, it provides little explicit structure for data concentrated near low-dimensional manifolds, where different regions of the data distribution may correspond to distinct local geometric or semantic factors. As a result, the reverse model must recover manifold-level structure almost entirely from an unstructured terminal reference distribution. We propose PTL-Diffusion, a proof-of-concept diffusion framework whose forward noising process converges to a nonconstant periodic family of Gaussian terminal laws rather than to a single invariant law. Unlike a phase-conditioned DDPM, where phase information only enters the denoising network while the forward process remains unchanged, PTL-Diffusion embeds phase structure directly into the forward noising dynamics. The proposed construction remains close to standard denoising diffusion models: for a periodically forced Ornstein--Uhlenbeck-type forward process, we derive closed-form forward marginals, the limiting periodic Gaussian terminal family, and explicit Gaussian reverse posteriors, enabling standard noise-prediction training. We also introduce an invariant-average regularization term coupling the phase-conditioned reverse dynamics through the averaged periodic reference law. Experiments on torus and cylinder point-cloud benchmarks and the Olivetti face dataset show that PTL-Diffusion improves manifold-level distributional matching over matched DDPM baselines, reducing phase-conditioned errors, feature-space covariance errors, and nearest-neighbour manifold distances. These results suggest structured terminal reference laws as a promising direction, while motivating more expressive phase constructions and larger-scale evaluations.

The framework of $α$-potential games has recently been introduced as a tool to analyze finite-player dynamic games, reducing the challenging task of finding approximate Nash equilibria to a control problem of minimizing a single function called $α$-potential function. In this work, we investigate the limiting behavior of $α$-potential games as the number of players $N$ tends to infinity. We show that potential mean field games (MFGs) arise naturally as this limit. Specifically, both the optimal values and the minimizers of normalized $N$-player $α_N$-potential functions converge to those of a mean field control (MFC) problem with measure-valued controls. We establish the equivalence of $\lim_{N\to\infty}α_N= 0$ with the existing conditions for potential MFGs, and provide an unified approach to construct the potential function for MFGs using the techniques from differential geometry in Wasserstein space. We further demonstrate that the objective of the limiting MFC problem serves as a potential function for the corresponding MFGs, an extension of the analogous finite-player setting. This connection yields new constructions of potential MFGs from a finite-player game, through the asymptotic condition $\lim_{N\to \infty}α_N= 0$. As a by-product, we establish propagation of chaos for $N$-player games converging to MFGs for general controlled diffusions with common noise and non-separable control interactions.

We investigate the high Mach number limit for the scaled compressible Navier--Stokes system in the critical Besov framework. In the scaled momentum equation, the pressure force is represented by the term \(\varepsilon^2\nabla a^\varepsilon\), where $\varepsilon$ is the inverse Mach number; as \(\varepsilon\to0\), the formal limiting system is the compressible pressureless Navier--Stokes system. The analysis is complicated by the absence of density dissipation in the limiting model and by the highest-order coupling created by the viscous terms. For \(d\geq2\), we prove the global well-posedness of the scaled system for small initial data and obtain estimates that are uniform with respect to $\varepsilon$. A crucial ingredient is a parameter-dependent lower-order estimate for \(\varepsilon a^\varepsilon\), which compensates for the purely transport nature of the density equation and allows the uniform bounds to be closed. Based on these estimates, we justify the high Mach number limit and recover a global strong solution to the pressureless Navier--Stokes system. For \(d\geq3\), we further derive quantitative error estimates between the scaled solutions and the pressureless limiting solution. More precisely, on each fixed finite time interval, if the initial discrepancy is of order \(\mathcal{O}(\varepsilon)\), then the corresponding lower-order critical Besov error satisfies the same rate, which yields a quantitative justification of the pressureless limit.

The spaces in the title are holomorphic function spaces on the unit disk containing a function with the bizarre property that, although it can be approximated in norm by polynomials, it cannot be approximated by the partial sums of its own Taylor series, nor by any linear combinations thereof. The search for concrete examples of such spaces leads to an elementary problem about convergent series that is still unsolved.

For graphs $F$ and $H$, the generalized Turán number $\mathrm{ex}(n,F,H)$ denotes the maximum number of copies of $F$ in an $H$-free graph on $n$ vertices. We prove that if $s\in \{2,3\}$, $s< a\leq b$ and $t$ is sufficiently large, then $\mathrm{ex}(n,K_{a,b},K_{s,t})=Θ(n^s)$. The $s=2$, $a=b=3$ case of this result answers a question of Spiro. Proving another conjecture of Spiro, we show that for every graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\mathrm{ex}(n,F,H)=Θ(n^r)$ holds for some graph $H$.

We generalize to the fibres of the fibration $\mathcal{O}\colon\mathbf{Cat}\rightarrow\mathbf{Set},$ defined by mapping a small category $\mathbb{X}$ to its set of objects $X_0=ob(\mathbb{X}),$ the classical notions of action and semidirect product of monoids. We prove that the equivalence between monoid actions of a monoid $Y$ and Schreier split extensions on $Y,$ which is well known to generalize the equivalence between actions and split extensions for groups, is an instance of a broader adjunction between Schreier points and actions in the fibres $\mathcal{O}^{-1}(B).$ This adjunction is an equivalence if and only if $B=1,$ i.e., for the category $\mathbf{Mon}$ of monoids. Similarly, we prove that there is an adjunction (which, in the case of monoids, results in a known equivalence due to Patchkoria) between Schreier internal categories in the fibres $\mathcal{O}^{-1}(B)$ and the category of crossed semimodules in $\mathcal{O}^{-1}(B).$ The latter are defined by translating in $\mathcal{O}^{-1}(B)$ the notion of crossed semimodule in $\mathbf{Mon}.$ Eventually, we prove that, by defining crossed modules appropriately, this last adjunction yields an equivalence between crossed modules and Schreier internal groupoids in the fibres of $\mathcal{O}.$

Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number M(n). Backtests at M(8) and M(9) calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision n=9 estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; n=11 and n=13 center-window estimates show a larger-contrast analogous pattern. The protocol estimate for M(10) is \[ \widehat M(10)=(8.9360\pm0.0010)\times 10^{78}, \] where the displayed uncertainty is the budget-based forecast scale from the cross-n scaling law under the production budget.

A biclique decomposition of a graph is a partition of its edges into complete bipartite subgraphs. We consider graphs whose vertices can be ordered such that the neighborhood of every vertex is the union of a sublinear number of intervals. We observe that these graphs admit compact representations in the form of biclique decompositions of small size. Here, the size of a decomposition is measured as the sum of the number of vertices of its bicliques. Combining this result with the existence of suitable vertex orderings for graphs of low neighborhood complexity, as proven by Welzl in 1988, we recover and extend several known results up to logarithmic factors. These results include upper bounds on the Zarankiewicz problem, matrix multiplication, quantum circuit complexity, and shortest path algorithms in ``well-structured'' instances.

We introduce a class of continuous time finite horizon mean field games where the objective function of the representative player depends on a hidden state, in addition to position, control, and the population distribution. While acting on the position dynamics, the agent has the option to pay for seeing the hidden state. We connect the original formulation of our model with a mean field model of optimal control with discretionary stopping, characterize solutions, and give a simple explicitly solvable example. For a class of $N$-player games with compatible information structure, we show that approximate Nash equilibria can be constructed starting from a solution to the limit model.

We propose a geometrization of compact metric spaces that is based on ideas of S. Banach and J. Mycielski. Then we prove the following conjecture of S. Banach and S. Ulam from 1935: in every compact metric space there exists a finitely additive probability measure, invariant under congruences. Moreover, our techniques allow us to solve a problem of M. Talagrand related to the Marczewski problem and the Banach-Tarski paradox with pieces having the property of Baire. We give also a very simple proof of the conjecture of Ulam about the product Lebesgue measure in the Hilbert cube and explain the existing results about congruence-invariant Borel measures in the language of our geometrization.

We describe the linearized dynamics near the Lamb-Chaplygin dipole, a classical traveling solution of the two-dimensional Euler equations. Exploiting the Hamiltonian structure of the system together with its symmetries, we identify all possible sources of linear instability. For general perturbations in $L^1\cap L^p$, $p>2$, growth can occur only through two explicit mechanisms triggered by: {\rm (i)} a nonzero circulation on the core of the dipole, and {\rm (ii)} a nontrivial component along the generalized eigenvectors associated with the eigenvalue $0$. In particular, we completely classify the spectrum and the Jordan chains of the operator associated with the linear dynamics. Both mechanisms hint for a nonlinear dynamics that may drift along the symmetry-generated family of traveling dipoles without moving away from it.

We propose and study all-multiplicity building blocks for tree-level string amplitudes in AdS. These are worldsheet integrals obtained by dressing the corresponding flat-space disc and sphere integrals with multivariable multiple polylogarithms and their single-valued analogues, respectively. We derive monodromy relations for the open-string building blocks and a KLT factorisation for their closed-string counterparts. This extends the non-commutative AdS uplift of lower-point flat-space structures to general $n$-point kinematics.

We study $\mathbb{Z}$-Ramsey ultrafilters, ultrafilters containing witnesses to every shift-invariant instance of Ramsey's theorem. We prove that it is consistent that there are no $\mathbb Z$-Ramsey ultrafilters. We also prove that every $(\mathbb{Z},3)$-Ramsey ultrafilter, as well as every $\mathbb{Z}$-Ramsey P-point, is selective. Further, we exhibit a generic extension -- using quotient algebras of the form \(\mathcal{P}(\mathbb{Z})/\mathcal{I}\) for certain \(F_{σ}\)-ideals -- that contains P-points that are not \(\mathbb{Z}\)-Ramsey ultrafilters, thereby addressing open questions raised by Petrenko and Protasov.

We introduce a family of homotopy theories for generalized metric spaces with natural number distances, via the magnitude-path spectral sequence (MPSS). The first page of the MPSS is known as magnitude homology; the second page is known as bigraded path homology, and contains GLMY path homology as its top row. For each natural number r, we define a class of maps of metric spaces called r-quasi-isomorphisms: those maps that induce a quasi-isomorphism at page r of the MPSS. We show that every page of the spectral sequence satisfies a suitable metric analogue of each of the Eilenberg-Steenrod axioms. In particular, we introduce the notion of r-cofibration and prove a Mayer-Vietoris theorem for page r with respect to r-cofibrations. We establish a family of Brown category structures on generalized metric spaces which allow us to explicitly compute homotopy colimits. We apply this to describe r-suspension and r-spheres of dimension n, and compute their spectral sequences. Finally we prove that for r = 1 the entire theory restricts to directed graphs.

The goal of this paper is to establish polynomial Fourier decay for images of self-similar measures $μ$ on $\mathbb{R}^k$ under sufficiently nonlinear real-analytic maps $f \colon \mathbb{R}^k \to \mathbb{R}^d$. For example, we prove that if $f$ is analytic on $\mathbb{R}^k$, its graph does not lie in an affine hyperplane in $\mathbb{R}^{k+d}$, and $μ$ is not supported in an affine hyperplane in $\mathbb{R}^k$, then the image measure has polynomial Fourier decay. Key steps in the proof include establishing a uniform Lojasiewicz-type inequality for self-similar measures, and using the decay of the Fourier transform of $μ$ outside a very small exceptional set of frequencies. As an application of our results, we prove polynomial Fourier decay for self-conformal measures on $\mathbb{C}$ for a large class of complex analytic IFSs which are not self-similar but are conjugate to a linear IFS via an analytic map.

Lovász' famous Hamiltonicity conjecture (1969) states that every connected vertex-transitive graph has a Hamiltonian path. A stronger version of the conjecture, often attributed to Thomassen (1978), states that every sufficiently large such graph even has a Hamiltonian cycle. Despite the great amount of attention these conjectures have attracted over the past decades both in the combinatorial and algebraic communities, for more than 40 years the best known lower bound for the maximum length of a cycle (path) in a connected vertex-transitive graph of order $n$ remained of the form $Ω(\sqrt{n})$, due to Babai (1979). A series of recent works has successively improved the exponent in this lower bound further. In this paper, improving the previous state-of-the-art bound $Ω(n^{9/14})$ due to Norin et al.~(2025), we prove that every connected vertex-transitive graph of order $n$ contains a cycle of length at least $n^{2/3-o(1)}$. This hits a natural barrier for several existing approaches from previous work. Our proofs combine recent embedding techniques for paths in sublinear expanders, sublinear expander decompositions of almost-regular graphs, and several additional combinatorial ideas.

In this paper, we study the monogeneity of a special class of composed polynomials of the form $ (f \circ g)(x) = (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b,$ where \( f(x) = x^n + a x^{n-1} + d x^{n-2} + b \in \mathbb{Z}[x] \) satisfies \( a^2 = 4d \) and \( g(x) = x^m + c \in \mathbb{Z}[x] \). Assuming that \( (f \circ g)(x) \) is irreducible over \( \mathbb{Q} \), we obtain necessary and sufficient conditions on the parameters \( a, b, c, d, m, n \) for the polynomial to be monogenic. These conditions help to identify when the set \( \{1, θ, \dots, θ^{mn-1}\} \) forms an integral basis of the number field \( \mathbb{Q}(θ) \), where \( θ\) is a root of \( (f \circ g)(x) \). We also provide lower bound for the counting of such monogenic polynomials. Furthermore, we study the behaviour of solutions to certain related differential equations and present a class of polynomials with non-square-free discriminants as an application of the main results.

We study an online version of the robust mean change point detection problem under a dynamic Huber contamination model with arbitrary contamination distribution and inlier distribution possessing exponentially- or polynomially-decaying tails. This robustness framework is systematically studied for the first time in the change point literature. For univariate data, we characterise the detection delay by partitioning the parameter space into four regimes, in terms of the true change location, signal size and contamination level. Efficient detection procedures are accompanied by matching lower bounds, up to poly-logarithmic factors. For the multivariate setting, we devise an efficient robust mean testing procedure and apply this to the robust online change point problem. The theoretical analysis of the robust mean testing procedure is the first in dealing with both Huber contamination and heavy-tailedness, and is thus of independent interest. Extensive numerical experiments are conducted to support our theoretical findings.

The triangle packing number $ν(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2ν(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random geometric graph for a large range of densities. We also study the problem of covering almost all edges of the random geometric graph with edge-disjoint copies of some fixed graph $F$. In particular, we show the existence of almost-perfect packings for an infinite family of $F$, and state some negative results as well.

For a clique $K_r$, a graph is $K_r$-saturated if it contains no copy of $K_r$ and the addition of any edge from its complement creates a $K_r$. A classical result of Erdős-Hajnal-Moon and Zykov shows that the number of edges of an $n$-vertex $K_r$-saturated graph is at least $(r-2)n-\binom{r-1}{2}$. In this paper, we focus on the number of edges of the $K_r$-saturated graphs with a fixed matching number. Let $G$ be an $n$-vertex $K_r$-saturated graph with matching number $ν(G) = s$. For sufficiently large $n$, we prove that the number of edges \begin{equation*} e(G)\geq \left\{\begin{array}{cl}{(r-1)n-\frac{r}{2}(r-1)-1,}&{\quad\mathrm{if}~s=r-1;}\\{(r-1)n + (s-r)^2 - \frac{1}{2}(r+2)(r-3) - 5,}&{\quad\mathrm{if}~s>r-1.}\\\end{array}\right. \end{equation*} Moreover, we completely characterize the graphs attaining the equality.

The second homology group is of central importance in the study of profinite rigidity of $3$-manifold groups. Although general and deep results imply that the integral homology of cocompact hyperbolic $3$-orbifold groups is computable in principle, the resulting algorithm is not practical. We develop an effective method for computing $H_2$ in the case of orbifold groups arising as finite extensions of the fundamental group of hyperbolic rational homology $3$-spheres. As a special case, this yields explicit computations of the second homology groups of all cocompact lattices between $π_1(\mathcal{W})$ and its normalizer in $\mathrm{PSL}_2(\mathbb C)$, where $\mathcal{W}$ is the Weeks manifold. We also show that these lattices are absolutely profinitely rigid, completing work by Bridson, McReynolds, Reid & Spitler in this setting. As a special case, we determine that $H_2(Γ_{\mathcal{O}}, \mathbb Z) \cong \mathbb{Z} / 2\mathbb{Z}$, where $Γ_{\mathcal{O}}$ is the normalizer of the group of units $Γ_{\mathcal{O}}^1$ in a choice of maximal order $\mathcal{O}$ of the quaternion algebra associated to $\mathcal{W}$, thereby answering a question of Bridson & Reid. Although this non-vanishing obstructs one possible construction of Grothendieck pairs in $Γ_{\mathcal{O}}^1 \times Γ_{\mathcal{O}}^1$, we use our computations to show the vanishing of the second homology of another lattice whose derived subgroup is $Γ_{\mathcal{O}}^1$, which then yields Grothendieck pairs in this direct product by a theorem of Bridson & Reid. Finally, to showcase the generality of the techniques, we also compute the second homology of some finite extensions by orientable isometries of the fundamental group of some Fibonacci manifolds $M_n$.

We study the problem of optimally sending a real-valued source through multiple uses of a channel with feedback. First, we state a set of conditions that are sufficient for an encoder to achieve maximal mutual information between the source and all the channel outputs. This set of conditions are also necessary when the channel is input-identifiable, a condition widely satisfied by common channel models. More notably, we further study the information maximization-distortion minimization problem, where the mutual information between the source and all channel outputs still needs to be maximized, while at each step, the MMSE of estimating the source from the channel outputs so far also needs to be minimized. We derive a solution to this problem for discrete channels with certain symmetries, e.g. $k$-ary symmetric or $k$-ary erasure channels. We show that for such channels the famous posterior matching scheme, while not necessary for information maximization alone, is sufficient and essentially necessary for achieving both information maximization and distortion minimization. This work also provides a new perspective of regularizing distortion-minimizing feedback communication through information maximization, which enables us to find the optimal solution that otherwise would be intractable.

Consider the set of isoenergetically degenerate integrable Hamiltonians with two degrees of freedom. We show that a cusp-generic perturbation of a generic Hamiltonian in this set gives rise to meandering invariant tori - embedded Lagrangian tori which are not graphs. Moreover, an exponentially dense subset of perturbations admits higher order meandering tori, of all orders from two to infinity. These infinite order meanders have an endless nested structure.

Field quantization is a central feature of modern physics, that underpins the concept of photons and forms the foundation of quantum electrodynamics as well as much of solid-state theory. Classical linear wave equations are not generally expected to reproduce the quantization arising in quantum systems without introducing additional ingredients such as ad hoc nonlinear constraints, resonant particle-wave couplings or stochastic background fields. Here, we show that appropriately engineered linear wave media can recover fundamental features evocative of energy quantization in quantum mechanics. The key is to tailor periodic media in which wave propagation is strongly suppressed, except over a discrete set of narrow pass bands. In this regime, stationary wave solutions exhibit discrete energy and frequency spectra analogous to those arising in quantum mechanics despite the underlying dynamics remaining linear. Owing to the universality of the proposed mechanism, these effects may be realized experimentally using mechanical, electrical, or electromagnetic waves in appropriately designed periodic media. This work opens new avenues for designing metamaterials that enable control over discrete wave states while strengthening the conceptual bridge between classical and quantum wave physics.

Let $X\subseteq \mathbb{P}^n_{\mathbb{F}_q}$ be a reduced, equidimensional, quasiprojective scheme. We prove that there exists a positive-density set of hypersurfaces $H_f$ such that for every closed point $P\in X\cap H_f$, one has $\mathrm{ord}_P(f)=1$ and $e_P(X\cap H_f)=e_P(X)$.

We study degenerate controlled diffusions and Hamilton--Jacobi--Bellman equations posed on genuine continuum percolation clusters. The diffusion is constrained to evolve inside the infinite cluster and degenerates according to the distance to the irregular random boundary. Under suitable regularity and degeneracy assumptions on the diffusion matrix, we prove that the corresponding controlled diffusion never reaches the boundary and therefore admits a unique global strong solution. Using this result, we establish a stochastic control representation formula for viscosity solutions of degenerate Hamilton--Jacobi--Bellman equations posed on the random cluster, without imposing boundary conditions. We further verify that the structural assumptions introduced in this work, including quantitative integrability properties of the distance-to-the-boundary function and the associated degeneracy, hold for concrete continuum percolation models such as the Boolean model. In particular, although the diffusion never reaches the boundary, the boundary geometry remains a fundamental ingredient of the theory through the admissible degeneracy regime. The results establish a quenched framework for stochastic control and degenerate partial differential equations on genuine continuum percolation geometries, and identify a link between the analytic structure of the diffusion and the stochastic geometry of the underlying cluster. They also provide the foundational framework for the homogenization theory developed in the companion article [BMM26].

We develop a numerical algorithm for the efficient solution or approximation of solutions to the Helmholtz equation on a structured grid in two dimensions. We make use of the Operator Fourier Transform (OFT) and a low-rank cross approximation scheme (Cross-DEIM) to decompose the problem into an integral over a pseudo-time of solutions to the Schrödinger equation. The OFT is a framework for solving operator equations like fractional Laplacian equations or the Helmholtz equation, when the latter is written as a product of two paraxial operators. The main computational cost in the OFT is the solution to the Schrödinger equation, especially when the dimension or mesh resolution is high. In this work, we alleviate this cost by utilizing a low-rank method. Such methods aim to beat the curse of dimensionality when low-rank structures are present in the solution. We show that the combination of these two approaches can have large cost reductions for certain classes of problems.