Papers

Let $I=(a,b)$ be an open interval with finite or infinite endpoints. For $1<p_1,p_2<\infty$ and $0<q<\infty$, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from $L^{p_1}(v_1;I)\times L^{p_2}(v_2;I)$ to $L^q(u;I)$, including the quasi-Banach target range $0<q<1$. With $\vartheta=\min\{1,q\}$, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the $\vartheta$-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for $q<1$ explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex $L^{1/2}(0,1)$ example for which $a_{m+1}^{(2)}(\mathcal H_2)=O(m^{-1/2})$.

The classical information bottleneck (IB) measures the relevance of a representation $U$ of $X$ to a target $Y$ by $I(U;Y)$, which does not directly characterize the error of downstream decisions. For a binary hypothesis $Y$ inferred from many separately encoded observations, the optimal error exponent is the Chernoff information between the two conditional distributions of $U$ given $Y$. We study the mutual information constrained Chernoff bottleneck, which seeks an encoder that maximizes this Chernoff information subject to a rate constraint $I(U;X) \leq R$. We show that its optimal value $C(R)$ increases strictly up to $R = H(V)$, where $V$ merges the symbols of $X$ with equal likelihood ratio, remains at the uncompressed exponent beyond, and, unlike the IB curve, need not be concave. We further show that $k+1$ outputs suffice to attain $C(R)$, where $k$ is the cardinality of $V$. We propose an alternating algorithm that updates the encoder via a generalized Blahut–Arimoto algorithm and the Chernoff parameter $s$ via a nonlinear equation, and prove that its iterates remain feasible, with nondecreasing and convergent Chernoff information. Comparing with the classical IB for $n$ samples, we show that, whenever the two objectives disagree, the IB encoder has the smaller equivocation $H(Y|U^{n})$ at $n = 1$, whereas the Chernoff bottleneck has the smaller error probability and equivocation for all large $n$. Numerical experiments confirm the theory, and on real topic-detection data from the 20 Newsgroups corpus, compressing each word to only $17\%$ of its entropy retains $90\%$ of the error exponent and nearly the accuracy of the uncompressed classifier.

The Erdős-Rogers function $f_{\ell,s}(n)$ is the largest order of a $K_\ell$-free induced subgraph guaranteed to exist in every $K_s$-free graph on $n$ vertices. While this function is well understood for $s=\ell+1$, the case where $s$ is much larger than $\ell$ has remained wide open. A long-standing lower bound of Sudakov states that $f_{\ell,s}(n)\geq n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$, while a recent result of Bradač shows that $f_{\ell,s}(n)\leq n^{\frac{\ell-1}{s-1}+o(1)}$. In this paper, we close this gap asymptotically by proving that $f_{\ell,s}(n)= n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$. More precisely, we prove that for all $2\leq \ell<s$, we have $f_{\ell,s}(n)\leq n^{\frac{\ell}{2s-\ell}+o(1)}$. Our proof builds on Bradač's recent tight construction for off-diagonal Ramsey numbers, which can be viewed as the $\ell=2$ case of our result.

An independent set of a graph G is a subset of VG, no two of which are adjacent. The cardinality of a maximum independent set in a graph G is called the independence number of G, denoted by alpha(G). The essential connectivity kappa'(G) of a graph G is denoted as the minimum number of vertices of G whose removal produces a disconnected graph with at least two non-trivial components. In this paper, we determine the n-vertex connected graphs with given independence number and essential connectivity that attain the minimum distance spectral radius, and fully characterize the corresponding extremal graphs.

We prove a sharp quadratic inequality between the largest two eigenvalues $\lambda_1 \ge \lambda_2$ of a graph with $n$ vertices. We also prove a quadratic inequality between the second and third largest eigenvalues $\lambda_2 \ge \lambda_3$. These results in particular imply the bounds $\lambda_1 + \lambda_2 \le \frac{8}{7} n - 2$, $\lambda_3 \le \frac{n}{3} - 1$ and $\lambda_2 + \lambda_3 \le \frac{2}{3} n - 2$. In fact we determine the closure of the set of possible $(\frac{\lambda_1+1}{n}, \frac{\lambda_2+1}{n}) \in \mathbb{R}^2$ and $(\frac{\lambda_2+1}{n}, \frac{\lambda_3+1}{n}) \in \mathbb{R}^2$. More generally, we prove quadratic bounds in the case of symmetric matrices in $[0,1]^{n \times n}$, and we also give a quadratic bound for two eigenvalues of a symmetric matrix in $[-1,1]^{n \times n}$. These bounds are proved by transforming the problem into extremal geometric questions in $\mathbb{R}^3$ and $\mathbb{R}^2$. We use the method of Lagrange multipliers to reduce to special cases with at most five points, and we deal with these special cases directly.

A graph is $K_3$-irregular if its vertices belong to pairwise distinct numbers of triangles. We prove that no $8$-regular $K_3$-irregular graph exists, settling the last unresolved case. Following the initial discovery of such graphs for regularities $r \in \{10,11,12\}$ (Stevanović et al., 2024), our previous work (Hak et al., 2025) showed that no such graphs exist for $r \le 7$, provided the first example for $r=9$, and proved that any $8$-regular candidate must have between $17$ and $22$ vertices. We exclude these possible orders for $r=8$ by combining careful analysis of triangle degrees with integer linear programming techniques. Meanwhile, a recent construction (Zhang, 2026) established that regular $K_3$-irregular graphs do exist for all $r \ge 9$. Together with our results, this establishes that an $r$-regular $K_3$-irregular graph exists if and only if $r\geq 9$.

Motivated by anisotropic viscosity in geophysical fluid dynamics, where vertical momentum diffusion is often much weaker than horizontal diffusion and may be negligible in the interior yet remains essential near a solid wall, we study the vanishing vertical viscosity limit for the two-dimensional incompressible Navier–Stokes equations in the upper half-plane, with horizontal viscosity fixed at one and vertical viscosity \(\varepsilon^2\). For arbitrary divergence-free \(H^4\) no-slip initial data, with no time-differentiated compatibility conditions required, we construct the limiting horizontally viscous flow and the boundary-layer profiles on every prescribed finite interval \([0,T]\). The leading-order corrected approximation has \(O(\varepsilon)\) error in \(L^\infty\), and the full finite-order expansion reduces this error to \(O(\varepsilon^{3/2})\), uniformly down to \(t=0\). We further describe the development of the layer from data that vanish initially at the wall. At the initial–boundary corner, we determine the first two self-similar coefficients and characterize matching through exact profile tails, thereby separating the fixed-time scale \(\varepsilon\) from the short-time scale \(\varepsilon\sqrt t\). When the initial wall acceleration is nonzero, both the leading correction rate and the first normalized corner remainder are sharp. Finally, continuity estimates for fixed data yield joint small-time and small-viscosity limits for the exact solution, as well as finite-\(L^p\) asymptotics.

Let $M$ be an imaginary quadratic field. Let $A$ be a polarised abelian threefold defined over $M$ with geometric endomorphism algebra isomorphic to $M$. We study residual Galois representations attached to $A$. In particular, we describe the endomorphism field and the natural restrictions placed on the residual representations by their endomorphisms. We then devise criteria for the image of residual Galois representations to be large, and provide efficient algorithms suitable for large scale calculations. Subsequently, we apply this algorithm to millions of curves in several families, whose Jacobians are abelian threefolds with imaginary multiplication, and to Sutherland's dataset of $7$-smooth Picard curves. We produce an explicit example of a Picard curve which appears to have an isogeny of degree $13$ defined over $\mathbb{Q}(\zeta_3)$. The Jacobians of all other curves, in the range of our computation with endomorphism algebra $M$, have mod-$\ell$ image as large as possible for any prime $\ell>7$. This allows us to realise the group $\Gamma\mathrm{U}_3(\ell)$ of unitary semisimilitudes as a Galois group over $\mathbb{Q}$ for all $\ell \not\equiv 1, 25, 121 \pmod{168}$. Our algorithms also led us to the discovery of several interesting rational families of Picard curves whose generic members appear to have endomorphism algebra of dimension $6$ which is not a CM field. These represent curves on the Picard modular surface and appear to be Shimura curves. Some of them appear to parametrise non-principally polarised abelian surfaces with quaternionic multiplication.

A majority edge-colouring of a graph $G$ is a colouring of the edges of $G$ such that, for every vertex $v$ of $G$, at most half of the edges incident with $v$ receive the same colour. This notion was introduced by Bock et al. in 2023, who proved that every graph of minimum degree at least $2$ has a majority $4$-edge-colouring. We investigate an on-line variant of majority edge-colouring in which the graph is revealed by the Presenter edge-by-edge and the Algorithm must colour each edge immediately and irrevocably. In particular. we prove that the greedy strategy, which uses at most five colours, is optimal among on-line algorithms if $\delta = O(\frac{\log n}{\log\log n})$. We further extend our results to $1/k$-majority edge-colourings.

Let $Q$ be a local ring, $f$ a nonzerodivisor, and $R=Q/(f)$. We characterize weak and Serre liftability of cyclic $R$-modules under suitable hypotheses, obtaining complete-intersection criteria for Serre liftability in small height. We prove that the tensor product of two weakly liftable cyclic modules remains weakly liftable when their first Tor module vanishes, and give sufficient conditions for tensor products to preserve Serre liftability. We construct an Artinian Gorenstein quotient $Q/I$, with $Q$ regular, that is perfect over $R$ and Serre liftable but not weakly liftable to $Q$, showing that the negative answer to a question of Jorgensen persists in this more restrictive setting. Finally, we characterize when a lift induces a lift of the Auslander transpose and obtain a freeness criterion related to the Auslander-Reiten conjecture, recovering a theorem of Ghosh and Samanta.

Motivated by the work of Andrews and El Bachraoui on partitions with repeated smallest parts, Malik and Sarma recently extended this concept to overpartitions. Using generating functions and $q$-series techniques, they established several results, including four identities, and posed the problem of finding combinatorial proofs for these identities. Recently, Baruah, Li, and Mohanta provided combinatorial proofs for three of these identities. In this paper, we present a combinatorial proof of the remaining identity.

Given an $n\times n$ symmetric matrix $M$ with largest eigenvalue $\lambda_1\geq 0$, it is easy to show that the solution of the optimisation problem $\max_{v\in [-1,1]^n}v^TMv$ is at most $\lambda_1 n$. We prove the following converse: if every $n'\times n'$ principal submatrix of $M$ has maximal eigenvalue at least $\lambda$, then $\max_{v\in [-1,1]^n}v^TMv\geq \lambda(n-n'+1)$. This allows us to lift results about the second largest and the smallest eigenvalue of graphs to the (a priori more difficult) setting of positive discrepancy and MaxCut. Using this new connection, we prove that every $n$-vertex $m$-edge graph that is far from a disjoint union of cliques has a cut of size at least $m/2+n^{5/4-o(1)}$, which is sharp up to the $o(1)$-term. Moreover, we prove that every $d$-regular $n$-vertex graph has bisection width at most $dn/4-\Omega_{\varepsilon}(d^{1/3}n)$ for $d\leq (1-\varepsilon)n/2$, which is optimal for $d=\Omega(n)$. This confirms a conjecture of Räty, Sudakov and Tomon.

We formulate a class of systems of Euclidean correlation functions for compact $d$-dimensional conformal field theories which are not necessarily unitary. We also introduce conformal $d$-vertex algebras, $d$-dimensional analogues of full vertex algebras, and prove that such an algebra is obtained from every system of conformally covariant correlation functions. Finally, for each $d\geq 2$, we construct a family of examples satisfying these axioms, corresponding to generalized free scalar fields. For $d>2$, this family contains both unitary and non-unitary examples.

We show that a closed spin four-manifold with nonzero signature, even after being punctured, does not admit a complete metric with uniformly positive scalar curvature. In particular, the punctured $K3$ surface does not admit such a metric. This is known to be false in higher dimensions. A key new observation, which could be of independent interest, is that Gromov's $\mu$-bubbles are a convenient class of boundaries for the APS boundary value problem for the Dirac operator.

Let $u:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if \begin{equation*} u = h \circ \pi \circ \Xi, \end{equation*} where $\Xi:\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}$ is an isometric transformation, $\pi:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ is the Hopf map and $h:\mathbb{CP}^n\to\mathbb{CP}^n$ is a holomorphic map with degree one. When $n=1$, it refines a classical result of Urakawa and a recent result of Rivière. Moreover, we prove that $h\circ\pi\circ \Xi$ has nullity $3n^2+5n$.

A wide range of methods have been proposed, including physics-informed neural networks, which are powerful but do not guarantee identifiability of the dynamics, symbolic regression, which requires a set of precomputed operations, and causal discovery, which is more principled but usually relies on strong assumptions that physical systems may violate. In this work, we develop a theory-grounded method and prove that under a set of permissive assumptions, the structural drivers and drift of stochastic delayed differential equations are identifiable. Our method outperforms others on a benchmark for driver identifiability, and on a second benchmark to evaluate physical consistency of the learned dynamics.

We establish convergence of minimizers for a minimal one dimensional atomic-scale model for relaxation in moiré materials to those of a one-dimensional continuum limit functional analogous to those commonly used in practice to model relaxation. The atomic-scale model consists of two periodic chains containing $N$ and $N+1$ atoms per common cell and is exactly moiré-periodic at the atomic scale. Its energy combines harmonic nearest-neighbor intralayer interactions with nonlocal interlayer interactions generated by an even $C^2$ pair potential with polynomial decay. The associated continuum energy couples linear elasticity to a periodic misfit potential, obtained by periodizing the pair potential, which penalizes unfavorable local stackings.

We introduce a novel proof strategy for semiparametric Bernstein-von Mises theorems based on Stein's method combined with an information-geometric framework. Rather than controlling the effect of the prior on the asymptotic marginal posterior distribution of the functional of interest through the stability of an integrated likelihood under a suitable perturbation, we characterise its influence through the prior-weighted divergence of suitable vector fields over the statistical model. Applying Stein's method in this setting instead of the usual Laplace-transform approach yields explicit non-asymptotic upper bounds on the bounded-Lipschitz distance between marginal posterior distributions and the corresponding Gaussian limits predicted by semiparametric efficiency theory. These bounds consist entirely of local scalar differential quantities associated with the functional, the prior and a chosen vector field-typically related to the efficient influence function-evaluated over posterior contraction sets. We apply the theory to quadratic functionals in Gaussian white-noise models and to linear, quadratic and general integral functionals in histogram density models, with both conjugate and non-conjugate priors. For linear and quadratic functionals, the same theory identifies the differential term responsible for posterior bias and allows us to remove it through a natural functional correction, yielding Bernstein-von Mises-type theorems in regimes where they may fail for the original uncorrected functional.

We compute the Dolbeault cohomology of minimally stable polytopal LVM manifolds using a finite curvature complex. Let $B$ be the indispensable weight block. A canonical exact sequence identifies the kernel of the curvature map with $(\operatorname{coker}B)^\vee$ and its cokernel with $(\ker B)^\vee$. At full rank, the Dolbeault groups are direct sums of reduced cohomology groups of induced subcomplexes of the visible simplicial sphere, tensored with the exterior algebra on the complex dual of the deck lattice. One proof uses the curvature model and Stanley–Reisner Tor; a second gives an additive comparison through completed Laurent expansions and holomorphic descent. Full curvature rank is equivalent to Frolicher degeneration at $E_1$. Every manifold in this class fails the $\partial\bar\partial$ lemma. For polygons, the number of facets and the curvature rank determine the entire Hodge diamond. We construct a proper holomorphic family over a disk with fixed visible square whose curvature rank drops at the origin. The total dimension of Dolbeault cohomology increases there by sixteen. At full rank, holomorphic descent also computes tangent-sheaf cohomology, including resonant contributions in positive Cech degree. We determine the Kodaira–Spencer map of the normalized weight family and give a cohomological criterion for semiuniversality with smooth base. For a resonant example over the square, the global vector fields have a polynomial basis of nine elements and $h^1(\Theta)=19$. The primary obstruction map is nonzero. The additional Cech class integrates in a holomorphic family.

We study g-dimonoids, which are algebraic structures equipped with two associative binary operations satisfying a prescribed system of axioms. We develop a construction of g-dimonoids based on inflations of semigroups, construct several classes of 2-dinilpotent commutative g-dimonoids, and investigate the isomorphism problem for these classes. As an application of these results, we obtain a structural description of all commutative g-dimonoids of order 3, up to isomorphism.

We implement a Macaulay2 package for computations involving rings of truncated Witt vectors $W_n(R)$ of a finitely generated $\F_p$-algebra $R$. This package includes ring operations (addition, multiplication, Frobenius, Verschiebung, etc.) on elements of $W_n(R)$, conversion between tuple representatives and ghost map representatives of elements of truncated Witt vectors over polynomial rings, and explicit computation of $W_n(R)$ as a finite-type algebra over $\Z/p^n$. This functionality is based on an algorithm we developed for performing arithmetic operations in rings of finite length $p$-typical Witt vectors over finitely generated $\F_p$-algebras. In addition, we implement an algorithm to calculate lifts of Frobenius from $\F_p$-algebras to flat lifts over $\Z/p^2\Z$, and an algorithm to find the quasi-$F$-splitting height of a local or graded complete intersection ring.

In 2018, Andrews defined and studied the partition function $\mathcal{EO}(n)$, which counts the number of partitions of $n$ where each even part is less than each odd part. Thereafter, Uncu considered a different subset of such partitions, namely those partitions counted by $\mathcal{EO}(n)$ where there are no repeated even parts (we denote the number of such partitions by $\mathcal{EO}_u(n)$). In this paper, we prove several congruences modulo powers of $2$ for $\mathcal{EO}_u(n)$. We also prove some infinite families of congruences as well as a recurrence for the number of this class of partitions. Our methods involve elementary, algorithmic, and modular form techniques.

Projection pursuit searches for a direction along which the data look least Gaussian. When the observation space contains a large Gaussian complement, the empirical objective can be minimized by a direction that carries no signal, with empirical kurtosis as low as at the truth. Sample splitting exposes rather than repairs this failure. Appending coordinates independent of the latent regime degrades the search while leaving Bayes recoverability unchanged. Restricting the search to the column space of a known forward operator removes the failure exactly on the negative-kurtosis branch. Estimating a principal subspace from the data is the alternative. In a controlled two-component model, the leading sufficient scalings differ in the gain with which the operator transmits the discriminant: $\varsigma^{-4}$ for covariance-spike estimation and $\varsigma^{-8}$ for fourth-moment search. At fixed search dimension, the measured threshold ratio collapses onto $n/p^2$ with exponent $0.156$, close to the predicted $1/8$. This is an empirically supported scaling motivated by sufficient bounds, not a proved asymptotically tight law. When the search dimension is varied, the measured exponent is $0.325$, substantially larger than $1/8$, and the tested range does not identify its functional form. The crossing location also depends on calibration and model configuration. Under a downstream excess-error criterion, the scaling largely disappears.

We show that low discrepancy is equivalent to spectral gap for directed $k$-uniform hypergraphs. For undirected hypergraphs this recovers a theorem of Lenz and Mubayi, with a considerably shorter proof. At the heart of our argument is a Frieze–Kannan type spectral regularity lemma for hypergraphs based on the variational notion of hypergraph eigenvalues by Friedman–Wigderson, which decomposes any tensor into a bounded number of rank-one tensors plus a quasi-random tensor with small top eigenvalue. This lemma may be of independent interest, and we prove it for complex-valued, not necessarily symmetric tensors. We also briefly discuss a Szemerédi-type variant and the regularization of Cayley-type hypergraphs over finite abelian groups.

We use multipoint semidefinite programming to prove the nonexistence of strongly regular graphs. A normalized eigenspace projection of a primitive strongly regular graph gives a spherical two-distance set with one point for each vertex. A bound smaller than the required number of points therefore rules out the graph. Following the formulations of de Laat et al. and Kao and Yu, we use, for each reference type, coefficient matrices indexed by the feasible inner-product labels of the selected points. In degree zero, we combine the matrix coordinates indexed by the reference points into a single coordinate. The resulting programs use configurations of up to six points, with matrix orders at most $2^m+1$ for a reference set of $m$ points. Feasible dual solutions, verified in exact rational arithmetic, exclude strongly regular graphs with parameters $(351,140,73,44)$, $(550,162,75,36)$, $(703,182,81,35)$ and $(1344,221,88,26)$.

We consider a gradient, speed-change, symmetric exclusion process on the discrete torus $T^d_n$, $d\le 3$, whose empirical measure evolves, on the diffusive scale, according to a non-linear parabolic equation. We prove that, starting from a sequence of initial states whose relative entropy with respect to the inhomogeneous product measure associated with the initial profile is $o(n^{d/2})$, and whose density fluctuation field converges, the density fluctuation field converges, in $L^1(0,T; H_{- r})$, to the solution of a linear stochastic partial differential equation with time-dependent coefficients driven by a conservative space-time white noise.

The aim of the paper is to study Euclidean and l-finite Euclidean modules over a ring, submodules of their finite direct sums, reduced and strong reduced echelon forms of matrices with coefficients in Euclidean modules and bimodules. The concept of Euclidean module is a module-theoretic analogue of the concept of Euclidean ring. The class of (not necessarily commutative) Euclidean rings is a nice but `very small' class of rings that have arithmetic-like properties. On the contrary, surprisingly, the class of Euclidean modules is a `large class' of modules. The Euclidean modules exist for many constructions of rings that are defined over arbitrary rings. For example, for every skew polynomial ring $A=D[x;\s, \d]$ with coefficients in an arbitrary ring $D$, the generalized Weyl algebra $D[x,y;\s, a]$, the Weyl algebra $A_1=K[x][\der; \frac{d}{dx}]$, $U(\sl2)$ and many classical algebras, there exist Euclidean modules.

We prove a conjecture of Smyth on the Mahler measure of non-reciprocal trinomials of height $1$, determining exactly when their Mahler measures are greater than or less than $M(x+y+1)=\rho=1.381356\ldots$, conjectured to be the smallest limit point of $M(P)$ for non-reciprocal polynomials. Conjecturally, our result gives the full spectrum of $M(P)<\rho$ for all non-reciprocal polynomials, not just for trinomials.

This work studies the relationships among sublinear valuation rules, uncertainty structures, and local specifications in a time-homogeneous Markovian framework with killing. These objects are linked, under finiteness and locality of the upper generator and a Lyapunov condition, by three maps: robust valuation, globalization, and localization. First, we show that the corresponding classes of uncertainty structures and sublinear valuation rules are order-isomorphic via the robust valuation map. Second, our analysis clarifies how local specifications constrain sublinear valuation and uncertainty structures, as well as what information localization and globalization preserve. In particular, the compositions of localization and globalization need not recover the original objects but yield canonical extremal elements. Third, the sublinear valuation generated by a local specification is characterized by the greatest viscosity subsolution of the associated Hamilton–Jacobi–Bellman equation. Finally, each uncertainty structure with killing admits a unique representation by a family of pairs consisting of a cumulative discounting process and an underlying state law. These results provide a framework for order relations, probabilistic and PDE-based representations, and model recovery in sublinear valuation without requiring uniqueness of the underlying martingale problems or a viscosity comparison principle.

A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols so that each symbol appears exactly once in every row and column. A partial Latin square of order $n$ is an $n \times n$ array whose cells are either empty or filled in such a way that each symbol appears at most once in every row and column, and at most $n$ distinct symbols are used. In 1983, Daykin and Häggkvist conjectured that every partial Latin square in which each row and column contains at most $n/4$ symbols, and each symbol is used at most $n/4$ times, can be completed to a Latin square. We prove that every partial Latin square in which each row and column contains at most $0.231n$ symbols, and each symbol is used at most $0.231n$ times, can be completed to a Latin square, significantly improving the previous best-known bound of $0.08n$, obtained by Fu and Weng. This problem can be seen as a partite analogue of the Nash-Williams conjecture concerning triangle decompositions of dense graphs, recently proved in the breakthrough work of Delcourt and Postle. Our proof uses a `discharging' strategy, adapting the approach of Delcourt and Postle, combined with a novel method to achieve a `balancedness' property, required for the partite setting.