Papers

We consider a continuous-time random walk on the discrete two dimensional box, driven by the discrete Gaussian Free Field (DGFF) acting as potential: When at a vertex, the walk waits an exponentially distributed time with mean given by the exponential of the field times an inverse temperature parameter and then jumps to one of its neighbors uniformly at random. We prove that when the temperature is below the critical value the walk exhibits “aging” at a range of pre-equilibrium time scales: Observed at any such time and then again after an additional time of the same order, there is a positive probability that the walk is found within finite distance from where it was before, with this probability given asymptotically by the Generalized Arcsine Law with a temperature-dependent parameter. We show that this is a consequence of an intricate trapping mechanism which localizes the walk for periods of time which increase with the age of the system, and describe the complex structure of the underlying trapping landscape, which is intimately related to the geometry of the near-extreme level-sets of the DGFF. Altogether, this work demonstrates for the first time an Arcsine-Law aging in the case of a spin-glass-type system with a logarithmically correlated potential, throughout its glassy phase, as predicted in the physic literature.

We estimate the fractional moments of the normalized mass assigned by the Critical 2D Stochastic Heat Flow to small balls. Our results also cover the discrete case corresponding to the 2D directed polymer model and provide estimates that are uniform in all parameters. One key takeaway of our results is that the vanishing of the fractional moments is completely governed by the divergence of the second moment. We use a quite robust method, by refining the change of measure argument and introducing a novel coarse-graining procedure, reducing the proof to essentially second moment estimates (in fact, we also provide sharp second moment estimates for directed polymers, of independent interest).

In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form $\mathcal{M}=\mathcal{P} \cap H$, where $\mathcal{P}$ denotes the set of all probability measures on an arbitrary measurable space $(\Omega,\mathcal{F})$ and $H$ is an affine set of signed measures on $(\Omega,\mathcal{F})$ with finite variation. We first give a precise measure-theoretic formulation of the heuristic that extreme measures have minimal support. We then connect this with the notion of minimality with respect to absolute continuity, and prove that points that dominate no other element of $\mathcal{M}$ are the only ones realizing the blow-up of a certain divergence map for any suitable $\phi$-divergence. Finally, considering a different class of $\phi$-divergences, we recover a characterization of the extreme points of $\mathcal{M}$ as strict local maximizers of $\phi$-divergences relative to any suitable dominating measures. We apply our result to recover and complement results from the literature in the context of finite spaces, sets of measures defined by integral constraints, multi-marginal couplings, and dominated sets of probability measures.

We study the Vlasov–Poisson–Boltzmann system on the three-dimensional torus under uniform shear flow. For Maxwell molecules with Grad's angular cutoff and sufficiently small shear rate, we consider perturbations around the spatially homogeneous self-similar profile of the sheared Boltzmann equation. In self-similar variables, we prove the global stability of this profile, including global existence and uniqueness, the exponential decay of nonzero-spatial derivatives in weighted $L^\infty$ spaces, and the uniform boundedness of the renormalized perturbation. The analysis combines a Caflisch's decomposition, Guo's $L^\infty$–$L^2$ estimates, macro–micro analysis, and a spectral study of the zero-frequency mode. We further derive a closed zero-frequency system for a renormalized total energy and suitable second-order moments, which yields a precise large-time description of the shear-induced energy growth.

We introduce the notions of horospherical limit point and big horospherical limit point in the projective measured foliation space for a subgroup of the mapping class group, which are analogous to these notions for hyperbolic spaces. We classify Teichmüller geodesic rays directed by projective measured foliations into conical, horospherical, big horospherical limit points, and show that these can be determined by the topological properties of the directing foliations: except for the case of minimal and uniquely ergodic projective measured foliation, whose corresponding ray can be either a conical limit point or a non-conical horospherical limit point.

Over-the-air computation (AirComp) enables low-latency wireless data aggregation, but its accuracy is limited by imperfect signal alignment over fading channels and receiver noise. Fully digital beamforming improves aggregation accuracy in multiple-input multiple-output (MIMO) AirComp systems but requires one radio-frequency (RF) chain per antenna. To reduce this hardware burden, we investigate microwave linear analog computer (MiLAC)-aided beamforming for MIMO AirComp. Under a lossless and reciprocal MiLAC model, we jointly optimize the transmit digital precoding matrices and the receive-side MiLAC aggregation matrix to minimize the mean squared error (MSE). An alternating optimization algorithm is developed, in which the precoding matrices are optimally updated using the Karush–Kuhn–Tucker conditions and bisection, while the resulting convex aggregation matrix subproblem is solved globally using projected gradient descent. Numerical results verify the algorithm's convergence and demonstrate that MiLAC-aided beamforming approaches the MSE performance of fully digital beamforming with substantially fewer RF chains and outperforms phase-shifter-based hybrid beamforming under the same RF-chain budget.

We study a viscous one-velocity Baer–Nunziato system for two barotropic compressible fluids in a bounded three-dimensional domain. In contrast with models in which the volume fraction is merely transported or determined by an algebraic equilibrium constraint, it satisfies a differential closure driven by the pressure gap between the phases. For arbitrary finite-energy initial data and adiabatic exponents $\gamma^\pm>1$, we construct global-in-time dissipative measure-valued solutions and prove dissipative measure-valued–strong uniqueness relative to any sufficiently regular solution with the same initial data. The principal difficulties are the singular and non-continuous behavior of the pressure-relaxation source at vanishing volume fractions, the associated concentration defects, and the lack of direct coercivity of the standard thermodynamic relative energy with respect to the volume fraction. These are resolved by an endpoint cutoff argument and an augmented relative energy coupled to the renormalized volume-fraction equation.

The existence of global classical solutions for reaction-diffusion systems arising from chemical reaction networks remains a major open problem in the deterministic setting, especially for reactions with high polynomial growth. We prove that a suitably chosen, physically motivated transport noise yields unique strong solutions for complex balanced chemical reaction networks that are global in time with arbitrarily high probability. These solutions possess paths in $C^{\theta}_t C^{\infty}_x$ for all $\theta<1/2$ and are, in particular, classical in space. Furthermore, we show that a suitable transport noise can enhance the dissipation of spatial fluctuations at an arbitrarily prescribed exponential rate. Our proofs rely on a combination of scaling-limit arguments, maximal $L^p(L^q)$-regularity, and entropy-entropy dissipation estimates.

Convolution identities and recursive formulas have long played a role in the theory of holomorphic modular forms and their applications. These have served both as striking formulas and as fundamentally useful tools for applications to combinatorics. Recently, there has been renewed interest in examples arising from non-holomorphic modular forms. These include the famous Hurwitz-Kronecker class number relations dating to 1885, and groundbreaking work of Imamoğlu, Raum, and Richter from 2014. Imamoğlu, Raum, and Richter developed a theory of holomorphic projection for products of (vector-valued) harmonic Maass forms and holomorphic modular forms to produce many such formulas. This was related shortly thereafter by Duncan, Griffin, and Ono to replicable-type functions in the sense of Conway and Norton, and such recursions played a key role in their proof of the Umbral Moonshine Conjecture. Here, we develop new results on holomorphic projections of such functions which is more convenient for many natural cases. Imamoğlu, Raum, and Richter's choice corresponds to the case in which the generalized Pell equation $m^2 - Dn^2 = N$ has a square value for $D$, which has only finitely many solutions for a given value of $N$. Our main results cover the cases of general $D$, in particular those for which the Pell equation has infinitely many solutions. As an application, we resolve a recent conjecture of the second author. More generally, our approach yields 18 convolution identities for mock theta functions, which we boil down to 5 identities that can directly be used to prove the others. In the appendices, we also show how these identities can be proven by more direct $q$-series methods; however, the key utility of the holomorphic projection formulas is that they give a tool to automatically discover and verify such formulas.

We establish the existence of an energy minimizer for a variational model of compressible magnetoelastic solids. The analysis is carried out in a new admissible class of deformations consisting of mappings of finite distortion, which extends previously available existence frameworks. A key ingredient is a compactness result under the critical integrability assumption on the outer distortion coefficient, which significantly weakens the regularity requirements imposed in earlier works. To obtain this result, we prove a diameter estimate for finite-distortion mappings satisfying the Ciarlet-Nečas condition and derive an open mapping theorem under the optimal integrability assumption, that is, the outer distortion is in $L^{n-1}$. This provides a partial positive result in the direction of the Iwaniec-Šverák conjecture and implies that admissible deformations are homeomorphisms. These topological and compactness properties allow us to apply the direct method of the calculus of variations and establish the existence of minimizers for compressible magnetoelastic solids within the admissible class of deformations consisting of mappings of finite distortions.

Say that a topological space $X$ has finite, respectively bounded, cliques iff every closed, irreflexive binary relation — equivalently, every closed, loop-free directed graph — on $X$ has cliques of finite, respectively bounded finite, size. Every compact space has bounded cliques. Having finite cliques implies limit-point compactness, and is implied by $\omega$-limit point compactness (equivalently, countable compactness). Thus, for $T_1$ spaces, having finite cliques is equivalent to countable compactness. Having bounded cliques is strictly weaker than compactness. Indeed, any space $X$ such that $X^{\omega}$ is countably compact has bounded cliques. However, we have found no example of a countably compact space having finite but unbounded cliques. The existence of such a space is the major open problem raised in this note.

Restricted assignment minimizes makespan, the busiest machine's load, when each job has one size and a set of allowed machines. Its tightest studied relaxation is the configuration linear program, which fractionally gives each machine whole job sets that fit. Its integrality gap, the worst-case ratio of integral optimum to relaxation value, is open in general. On two-weight graph balancing (sizes 1 and 2, each job allowed on at most two machines) the gap is known: Jansen, Land, and Maack (2016) give both bounds. Their Table 1 instance attains 3/2, and their Corollary 11 bound of 2-s/b for sizes s < b matches it at {1,2}. We ask how small such an instance, a witness, can be. We give a six-job witness I* on the complete graph on four machines: unit jobs form a cycle through all four, and two weight-2 jobs take the remaining edges. I* has integral optimum 3 against relaxation value 2, one job fewer than the smallest earlier witness of this class. We prove I* minimum and unique at its size, on any number of machines: no instance with at most five jobs reaches gap 3/2, and at six jobs I* is the only witness, up to relabeling machines and adding unusable machines. At seven jobs uniqueness fails: we classify all thirteen witnesses, including the Jansen-Land-Maack instance. Eight jobs give 154, nine 1,662. At least four machines are necessary. Kartak et al. (2015) classified the extremal non-round-up instances of the same relaxation in one-dimensional cutting stock for small demand; the Discussion relates them. Recognizing witnesses at relaxation value 2, shared by all ours, is coNP-complete, so no min-max characterization with polynomially checkable conditions exists unless NP = coNP. Every feasibility decision behind the exhaustive claims was made twice, in floating-point and exact rational arithmetic, with identical results. We accept the pipeline's negatives only after it rediscovers I*.

In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. These operators are studied on Sobolev spaces with power weights measuring the distance to the boundary. We prove optimal estimates for the resolvent operators and the corresponding heat semigroups. In addition, it is proved that the Dirichlet and Neumann Laplacians admit a bounded $H^\infty$-functional calculus on Sobolev spaces with certain compatibility conditions at the boundary. We show that these compatibility conditions cannot be omitted in general. The results in this paper are a direct extension of those obtained by Lindemulder, Lorist, the author, and Veraar in [J. Funct. Anal., 289(8):110985, 2025].

We establish a criterion for the existence of a local Minkowski unit at $p$ that applies to all Galois extensions with Galois group isomorphic to the direct product of a non-$p$-group and a cyclic $p$-group. As applications, we construct non-abelian extensions admitting a local Minkowski unit at $p$ under various ramification conditions and analyze the Iwasawa module structure of units in $\mathbb{Z}_p$-extensions of number fields. We also extend our study to the case where the group of $p$-power roots of unity is nontrivial.

For any given time lag $\gamma\in(0,1)$, we prove that the one-sided parabolic BMO space $\mathrm{BMO}^+(\gamma)$ coincides with the parabolic BMO space $\mathrm{PBMO}^-(\gamma)$ with equivalent norms, the parabolic Muckenhoupt class $A_{\infty}^+(\gamma)$ defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes $A_r^+(\gamma)$ with $r\in[1,\infty)$, and the parabolic reverse Hölder classes $\bigcup_{q\in(1,\infty]}RH_q^+$ coincide with the parabolic Muckenhoupt classes $\bigcup_{r\in[1,\infty)}A_r^+(\gamma)$, and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016)]. To show them, we establish the uniform parabolic space-time shifting property for parabolic reverse Hölder weights, and develop the one-sided stopping time argument which yields a new parabolic John–Nirenberg inequality for $\mathrm{BMO}^+(\gamma)$. As applications, we obtain John–Nirenberg and exponential integrability characterizations of $\mathrm{BMO}^+(\gamma)$, prove that $\mathrm{BMO}^+(\gamma)$ is independent of the positive time lag, and identify its null space.

Input-to-state stability (ISS) provides a useful tool for analyzing the robustness of time-varying chemical reaction networks (CRNs). This paper investigates the ISS property for CRNs in two cases: one is for weakly reversible CRNs with nonzero deficiency and multiple linkage classes, which is beyond the limit of existing results requesting a weakly reversible CRN to be zero-deficiency and single-linkage-class; the other is for non-weakly reversible CRNs by leveraging the notion of linear conjugacy. These results, on the one hand enrich the studies of the ISS property of broader classes of CRNs, on the other hand serve for designing CRN-based molecular computing systems. The latter application suggests the ISS theory can justify the parallel mechanism of biomolecular computations. We use some CRNs of practical relevance to demonstrate our results, including the p53 signaling and dimerization network, the p21-activated kinase 1 network, etc.

In this paper, we consider the asymptotic density of $\#\{p\leq x:P^+(p-1)\geq p^c\}$ and $\#\{n\leq x:P^+(n)<P^+(n+1)\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, one has \begin{align*} \mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq\frac{16}{5}\log\frac{1}{c} \end{align*} where $e^{-5/16}<c<1$, and \begin{align*} \#\{n\leq x:P^+(n)<P^+(n+1)\}>0.299x. \end{align*} The first result constitutes an improvement upon that of Ding and Wang (2025), who obatined $\mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq \frac{7}{2}\log\frac{1}{c}$. The second result improves a previous result $0.280$ by the author (2026). The key to the proof is that for a special class of convolution forms equipped with well-factorable weights, we may use the level \(x^{5/8-o(1)}\) for Pascadi's prime-distribution result with triple-well-factorable weights. We also use Pascadi's estimation of incomplete Kloosterman sums.

We propose a stabilized sequential quadratic programming (SQP) method for degenerate constrained optimization problems on Riemannian manifolds. The problem considered in this study is a Riemannian nonlinear programming problem (RNLP) with equality and inequality constraints, where classical constraint qualifications may fail. While existing Riemannian SQP methods guarantee global convergence only under constraint qualifications, their convergence behavior is not ensured for degenerate problems. To address this limitation, we extend the stabilized SQP framework from Euclidean spaces to Riemannian manifolds. Without assuming any constraint qualification, we prove that the generated sequence has an accumulation point that is a Karush–Kuhn–Tucker (KKT) point, an approximate KKT (AKKT) point, or a stationary point of an associated feasibility problem. Finally, we conduct numerical experiments to confirm the effectiveness of the proposed method for degenerate problems.

The classes of SOP$_2$ and SOP$_3$ first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.

In this paper, we establish new fixed point theorems for multivalued mappings satisfying Hardy-Rogers type contractive conditions in complete $b$-metric spaces. Motivated by the constant-coefficient framework of Chaib et al. (Fixed Point Theory 27 (2026), 209-228), we develop a distance-dependent formulation in which the contractive coefficients are nonnegative functions of the interpoint distance. We introduce a piecewise contraction factor corresponding to the two cases determined by the relative sizes of the cross terms in the Hardy-Rogers inequality, thereby providing a formulation adapted to the structure of the contractive condition. Under appropriate assumptions on the coefficient functions, we establish fixed point existence through either $h$-upper semicontinuity or asymptotic limsup conditions. The resulting framework yields corresponding fixed point theorems for multivalued Reich, Kannan, and Chatterjea-type contractions. We further investigate strict fixed points and obtain conditions under which the fixed point and strict fixed point sets coincide and consist of a unique point. In addition, we establish a fixed point transfer result for suitable finite iterates, showing that the contractive condition may be imposed on $T^p$ rather than on the original mapping, provided that the latter has a strict fixed point. Corresponding single-valued consequences are also derived. Several examples illustrate the applicability of the results and demonstrate situations in which the proposed conditions apply beyond the corresponding existing formulations.

Symplectically self-polar bodies have been introduced recently. They appeared in the context of Mahler's conjecture and in the context of symplectic outer billiard dynamics. In this paper we discuss the topology of the space of symplectically self-polar convex bodies. Namely, we show that this space is contractible.

We construct three derived endofunctors on the derived category of graded integrable representations of the current algebra of a flat degeneration of $D(2|1,\alpha)$, and prove that their classes in the Grothendieck group are the genus two Macdonald operators. Computing the graded $\operatorname{Ext}$ pairing explicitly, we deduce the self-adjointness of the genus two Macdonald operators and the orthogonality of genus $2$ Macdonald polynomials with respect to the certain skew-bilinear form.

The ring $R_m$ of $m$-symmetric functions consists of the formal power series that are symmetric in the variables $x_{m+1},x_{m+2},\dots$ but carry no symmetry in the first $m$ variables. We develop a combinatorial theory for the specialization at $t=0$ of the Schur functions of $R_m$. Our main tool is a new characterization of right key tableaux as suprema of the sets of decreasing subwords of the reading words of the subtableaux of $T$. Being invariant under elementary Knuth transformations, this characterization is compatible with the RSK correspondence. We obtain in this way a generating function over semistandard tableaux for the $m$-symmetric Schur functions at $t=0$, together with a combinatorial proof of a Cauchy identity in $R_m$. The $m$-symmetric Schur functions and their dual are then respectively identified with Demazure atoms and Demazure characters. Restricted to the last $m$ variables, our correspondence specializes to a proof, by ordinary RSK, of Lascoux's nonsymmetric Cauchy identity for Demazure characters and atoms. As further applications, we relate the $m$-symmetric Schur functions at $t=0$ to the almost symmetric Schur functions through a unitriangular change-of-basis matrix, obtain tableau generating functions and Cauchy identities for both families, and derive Jacobi-Trudi type determinantal formulas for three different bases.

For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ \Sigma_0(M_k)=\frac{\pi^2}{6},\qquad \Sigma_j(M_k)=\Sigma_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently, Yang's monotonicity conjecture holds for the family $\{M_k:k\ge2\}$. We also show that the nonzero spectral data of the core operator are independent of $k$, whereas the numerical invariant sequence recovers $k$ from the length of its constant blocks. Thus the higher numerical invariants detect module-theoretic information invisible to the core spectrum.

This study applies finite automata to the automatic solving of a variety of combinatorial games. For games whose positions and moves can be represented as regular languages and their operations, we design a two-stage automatic solving algorithm: first, construct a candidate finite automaton to determine the $\mathcal{P}$- and $\mathcal{N}$-positions, and then perform rigorous formal verification on this automaton; once verified, a complete solution of the game is obtained. For partizan octal games, we introduce a generalized misère quotient, overcoming the limitation that traditional theory applies only to impartial games. Using the above algorithm, we successfully solve the majority of two-digit partizan octal games, and based on these results, we propose a partizan version of Guy's conjecture. We also successfully solve a considerable number of partizan octal games under misère play, and give a conjecture on the structure of those games exhibiting “algebraic periodicity” among them. For Kotzig's nim, we resolve the most important related conjecture: we prove that the outcomes and SG values are periodic under both normal and misère play (including their partizan versions). Our algorithm successfully solves several small-scale cases, including misère play and partizan versions. This study pioneers a new theoretical tool and algorithmic paradigm for the automatic solving of combinatorial games, and has broad prospects for further extension and application in the field of combinatorial game theory.

We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. We also consider applications to sign patterns in the Liouville function.

We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $\zeta_n:=\arccos(-\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\mathbb{S}^n$. We prove that $$ d_{\mathrm{GH}}(\mathbb{S}^n,\mathbb{S}^{n+1})=\frac{\zeta_n}{2} \qquad(n\geq1), $$ resolving a conjecture of Lim, Mémoli, and Smith. All cases $n\geq4$ were previously open. This equality is established by explicitly constructing a family of correspondences $\mathcal R_n\subseteq \mathbb{S}^{n+1}\times \mathbb{S}^n$, whose distortion matches the known quantitative Borsuk-Ulam lower bound $\zeta_n$. We also introduce synchronized spherical joins and suspensions of correspondences and prove that the distortion of a join is exactly the maximum of the distortions of its factors. In particular, suspension preserves distortion. Applying these join and suspension operations to the optimal correspondences $\mathcal R_n$ yields new bounds for spheres of nonconsecutive dimensions, including $$ \lim_{m\to\infty} d_{\mathrm{GH}}\bigl(\mathbb{S}^m,\mathbb{S}^{m+d(m)}\bigr) = \frac{\pi}{4} \qquad\text{whenever } d(m)\geq1,\ \text{and }d(m)=o(m).$$

Fluid antenna arrays (FAAs) reconfigure a finite set of radiating ports within a prescribed aperture. In compact apertures, however, channel-driven placement may cluster ports, strengthen mutual coupling, degrade radiation conditioning, increase source-voltage demand, and produce uneven current loading. This paper studies downlink multi-user beamforming with jointly optimized port placement and current-domain transmission. An electromagnetic-guided graph network predicts port layouts from channel observations and refines them using geometric and mutual-impedance information. The training objective jointly considers communication performance and electromagnetic feasibility, while a common evaluation procedure is applied to all methods. The results show that, under a common feasibility standard, the proposed method provides a controllable tradeoff among communication rate, current loading, and configuration latency.

Semantic communication (SemCom) and task-oriented communication (TOC) can reduce wireless resource consumption by focusing on transmitting semantic or task-relevant information instead of raw messages. In practice, a main challenge is to make transmitting information robust to channel noise and fading while keeping it compact. Existing learning-based transceivers often improve reliability by using larger encoders or higher-dimensional channel features, which increase computation complexity and channel uses. Therefore, optimized system design needs explicit rate control to balance performance and transmitting resources e.g., bandwidth and power. For this purpose, we propose a manifold-constrained hyper-connection (mHC) coding scheme with an entropy bottleneck (EB) for resource-efficient SemCom and TOC over wireless channels. Instead of using a single residual path of existing encoders, the proposed mHC-based semantic encoder applies multiple residual streams and constrains their interaction by doubly stochastic (DS) mixing matrices. The new structure improves representation diversity and training stability with negligible parameter and floating-point overhead. The EB quantizes the channel features and estimates the entropy-coded rate, enabling end-to-end rate–distortion/task optimization under bandwidth and transmit-power constraints. We further show that DS-constrained stream mixing does not increase the differential entropy of the transmitted features. This implies no increase in the ideal EB coding length. Experiments on SemCom and TOC under additive white Gaussian noise (AWGN), Rayleigh fading, Rician fading, and imperfect channel state information (CSI) show that the proposed scheme improves semantic/task performance, communication robustness, and convergence stability over residual and unconstrained HC baselines, while requiring no additional channel uses.

For an integer $n\ge0$, let $f(n)$ be the minimum number of subcubes of $\mathbb{Z}_3^n$ of the form $A_1\times\cdots\times A_n$, where $|A_i|=2$ for every $i$, whose union covers $\mathbb{Z}_3^n$. A simple counting argument gives $f(n)\ge(3/2)^n$, while $f(n)=O(n(3/2)^n)$ by random construction. We prove that $f(n)\le2(3/2)^n-1$, answering a problem of Imre Leader. We also show that $f(n)/(3/2)^n$ is nondecreasing and there exists a constant $C_3$ such that $f(n)=(C_3+o(1))(3/2)^n$ where $1.62227<C_3\le2$.