Papers

The commutators of intrinsic square functions associated with the intrinsic Littlewood–Paley $g$-function, the intrinsic $g_{\lambda}^{*}$-function and the intrinsic Lusin area function can be used to characterize BMO-type function spaces through their boundedness properties. In this paper, we show that, for $b\in {\rm BMO}(\mathbb{R}^{n})$, the boundedness of each of these commutators from the Musielak–Orlicz Hardy space $H^{\varphi}(\mathbb{R}^{n})$ to $L^{\varphi}(\mathbb{R}^{n})$ is equivalent to $b\in {\rm BMO}_{\varphi}(\mathbb{R}^{n})$ (a nontrivial subspace of $\rm{BMO}(\mathbb{R}^{n})$), under suitable assumptions on the growth function $\varphi$. This extends the weighted characterization of Han and Wu [Proc. Amer. Math. Soc. 152(1) (2024), 281–293] to the Musielak–Orlicz setting. In addition, under suitable assumptions on a growth function $\Phi$, we prove that, for $b\in {\rm BMO}_{\Phi}(\mathbb{R}^{n})$, these commutators are bounded from the $b$-adapted Orlicz–Hardy space $H_{b}^{\Phi}(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$ and from Orlicz Hardy space $H^{\Phi}(\mathbb{R}^{n})$ to $L^{1,\infty}(\mathbb{R}^{n})$.

We study Howard policy iteration for finite-horizon stochastic control problems in which the control enters the diffusion coefficient. Each iteration selects a policy by maximizing the Hamiltonian and evaluates it through a linear parabolic equation. Since the improvement depends on the Hessian, even smooth data can produce discontinuous evaluation coefficients. We prove global convergence in the full parabolic Sobolev space under assumptions that allow this loss of regularity. In one space dimension, bounded measurable coefficients and uniform ellipticity suffice, without a large discount, a short horizon, small diffusion variation, or regularity assumptions on the improving policies. Multidimensional extensions hold under structural conditions on the diffusion matrix. The proof combines monotonicity with a parabolic growth estimate to show that the Bellman residual vanishes in measure. Uniform higher integrability then yields strong convergence of the values, gradients, Hessians, and time derivatives, and constructs the unique strong solution. We also establish residual error bounds, extend the argument to entropy regularization and zero-temperature limits, and obtain quadratic convergence for special one-dimensional models.

Asynchronous iterative methods are attractive for large-scale parallel computing because they reduce synchronization and communication overhead. Existing convergence-rate analyses, however, have primarily focused on shared memory implementations, whereas distributed memory systems introduce more general and potentially inconsistent communication delays. In this work, we revisit asynchronous Jacobi and randomized Gauss–Seidel (RGS) methods for symmetric positive definite linear systems from a unified perspective. We first introduce a general asynchronous model that encompasses both shared memory and distributed memory settings and expresses the two methods through a common coordinate-update framework. We then establish linear convergence in expectation under an explicit stability condition. The resulting convergence bound depends algebraically on the delay through the quantity $\sqrt{\rho\tau}+\rho\tau$, where $\tau$ is the maximum communication delay and $\rho$ reflects the communication pattern of the underlying parallel implementation. In particular, under an appropriate scaling regime with $\rho\tau=O(1)$, the guaranteed per-iteration convergence rate has the same asymptotic order as that of synchronous RGS. These results provide a unified framework for quantifying the effect of asynchronicity on asynchronous Jacobi/RGS methods in both shared and distributed memory environments.

Let $N\subset \mathcal{T}_{g,n}$ be a connected complex totally geodesic submanifold. We prove that if $\dim_{\mathbb{C}}N\ge3$, then $N$ is the image of an entire Teichmüller space under a covering construction induced by a finite branched cover of marked surfaces. This answers a question of Arana-Herrera and Wright.

This paper grew out of an effort to understand the so-called uniform flatness, an important concept in the Bargmann–Fock interpolation theory developed by Varolin et al. We show that the graph of every polynomial in one complex variable is a uniformly flat algebraic curve in $\mathbb C^2$, with vanishing upper density with respect to the Gaussian weight. Combined with a result of Pingali and Varolin, this implies that such graphs are interpolating for the Bargmann–Fock space on $\mathbb C^2$.

Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erdős, Graham, Ivić and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n) > (1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a factor of order $\log_{3} n$, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes. The two problems are closely related, but a long prime-free interval does not by itself produce a large value of $K(n)$: what is needed is a weighted variant of the Erdős–Rankin construction. We follow the method of Erdős, Graham, Ivić and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered. This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.

We show that every planar graph with no $t \times t$ grid minor has treewidth at most $4t +4$. This improves on the previously best known bound of $\frac{9}{2}t - \frac{11}{2}$, due to Gu and Tamaki (2012), and is within a factor $2$ of optimal. A key step in the proof is showing the following result, which might be of independent interest: Every $2$-connected plane graph $G$ with radius $d$ and faces of size at most $k$ has a tree-decomposition of width at most $\max\{3d+ k+5, 2d+2k+1\}$ such that the vertex set of every face of $G$ is contained in some bag.

We study high-dimensional nonparametric regression on the Boolean hypercube \(\{0,1\}^d\), from a sample of size \(n\), under a Fourier–Sobolev budget, i.e., a bound \(B\) on the total quadratic influence of the regression function. The central object is the resolvable Fourier degree \(D^*(d,n)\), defined as the largest interaction order \(D\) such that the dimension of the degree-\(D\) Fourier–Walsh space does not exceed \(n\). We prove that the minimax risk over Fourier–Sobolev ellipsoids is of order \(B/D^*(d,n)\): estimation is possible up to the statistically visible interaction order, and the budget controls the residual mass carried by higher-order interactions. We then study monotone regression under the same budget. For monotone functions supported on a known set of \(s\) coordinates, with \(s\) of order \(\log n\), we obtain a matching minimax rate of order \(B/s\). Finally, we consider an unknown-support model in which the function is monotone and depends on at most \(s\) coordinates; its minimax risk is governed by the intrinsic dimension \(s\) together with the logarithmic cost of selecting the active support.

On a connected compact Kähler manifold without boundary, we derive a uniform upper bound for each Laplace eigenvalue over all Kähler metrics in a given Kähler class. This provides an affirmative answer to a question of Yau.

We study regression with subsets as covariates. The response is an unknown function of the input subset, and observations consist of noisy evaluations at uniformly sampled subsets, each containing exactly \(k\) items from a ground set of size \(d\). This problem arises in combination screening, bundle preference modeling, and other settings in which outcomes depend on collections of prescribed size. The fixed-cardinality constraint couples the membership coordinates, so standard product-domain notions of interaction and smoothness cannot be imported unchanged. We use the harmonic decomposition of the Johnson graph to define an intrinsic Johnson–Sobolev scale and determine the exact dimensions of low-order interaction spaces. We derive finite-sample risk bounds for harmonic projection and establish a nonasymptotic minimax characterization when the Johnson–Sobolev energy budget is controlled relative to the noise variance. We also show that stable least squares removes the signal-dependent fluctuation of empirical projection when the empirical Gram matrix is sufficiently well conditioned. For arbitrary signal-to-noise ratios, a rank-deficiency analysis quantifies the components left unidentified by the random design. Together, the rank-deficiency analysis and a centered completion estimator yield minimax bounds that match up to constants whenever \(\min(k,d-k)\) and the smoothness order are fixed. Numerical experiments reported in the Supplementary Material illustrate estimation under different interaction profiles and distinguish the effects of observation noise, random-design fluctuation, and incomplete coverage.

We construct a finite connected simple cubic graph $F$ such that every signing of its edges yields a signed adjacency matrix with an eigenvalue outside $[-2\sqrt2,2\sqrt2]$. This disproves the Bilu–Linial signing conjecture for general regular graphs. The graph $F$ is not Ramanujan, and the conjecture restricted to Ramanujan base graphs remains open.

We study the algebraic independence of special values of power series $f(\beta^{-1})$, where $\beta$ is a fixed Pisot or Salem number. In particular, we consider the case where $f(X)=\sum_{n\geq 0} t(n) X^{w(n)}$ is not a lacunary series and is not assumed to satisfy any special functional equation, such as a Mahler-type functional equation. In our main results, we give a new criterion of the algebraic independence of three values. Applying our main results, we prove that the following three values are algebraically independent: \[ % \begin{gathered} \sum_{n=3}^{\infty}\lfloor n^{y}\rfloor\beta^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=3}^{\infty}\beta^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=1}^{\infty}\beta^{-\lfloor n^{\log n}\rfloor}, % \end{gathered} \] where $y$ is an arbitrary positive real number. Since our criterion is flexible, we have considerable freedom in choosing the coefficients $(t(n))_{n\geq 0}$ and the exponents $(w(n))_{n\geq 0}$.

We exhibit smooth polarized surfaces $(Y,H)$ over $\mathbf{C}$ carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal $\mathbb{N}$-graded $\mathbf{C}$-domains, with an isolated singularity at the vertex, admitting no nonzero finitely generated graded maximal Cohen-Macaulay module. To the author's knowledge no such ring was previously known. Their existence contrasts with the theorem of Hartshorne, Hochster and Peskine-Szpiro that a three-dimensional $\mathbb{N}$-graded domain over a perfect field of characteristic $p>0$ always has one. As a consequence, in characteristic zero Hochster's small Cohen-Macaulay conjecture admits no graded refinement. The nonexistence criterion is numerical: for $|H|$ base-point-free with finite morphism, $H^2<K_Y^2-8\chi(\mathcal{O}_Y)=\tau(Y)$, the signature of $Y$ over $\mathbf{C}$, leaves no nonzero $\mathcal{E}$ with $H^1(Y,\mathcal{E}(tH))=0$ for all $t$. For Ulrich bundles, being semistable, this is Bogomolov's inequality; what is new is that no stability hypothesis and no condition on the Hilbert polynomial are needed, the Harder-Narasimhan filtration of $\mathcal{E}$ being compared instead with the Horrocks splitting of its direct image on $\mathbb{P}^2$. Hirzebruch's Hesse surfaces with $H_n=4A_n-E_n$, $n\ge3$, qualify. In dimension $m$ the obstruction reads $(m+1)H^m\ge(K_X^2-2c_2(X))\cdot H^{m-2}$, the inequality Lopez proved for Ulrich bundles; products and complete intersections inside them give examples in every dimension $\ge3$, and the threefold examples show that the characteristic-$p$ hypothesis in the criterion of Shimomoto-Tavanfar is essential.

We extend the Ibragimov-Khasminskii likelihood ratio analysis method, in the one-dimensional parameter case, to a framework based on the Skorokhod $M\_1$ topology. The proposed framework applies to a broad class of statistical models, including those for which the normalized likelihood ratio processes are continuous while the limiting likelihood ratio process is discontinuous, a setting outside the scope of classical approaches based on the uniform or Skorokhod $J\_1$ topologies. We derive sufficient conditions for the weak convergence of likelihood ratio processes in the space of càdlàg functions on $\RR$ vanishing at $\pm\infty$, endowed with the Skorokhod $M\_1$ topology, and show how this convergence yields the asymptotic behavior of the maximum likelihood estimator. We also introduce new techniques for controlling the $M\_1$ modulus of continuity. Although these techniques are particularly well suited to models in which all jumps and steep continuous transitions are in the same direction, they are of independent interest and may prove useful beyond this setting. As an application, we study a smooth change-point model for inhomogeneous Poisson processes in the fast regime, where the transition interval shrinks faster than $1/n$. We establish that in this case the maximum likelihood estimator has the same asymptotic behavior (consistency, rate of convergence, limiting distribution and convergence of moments) as in the corresponding ”pure” change-point model.

Suppose \(a, b, c\) are polynomials with \(ab \neq 0\) and \(c\) nonconstant. We determine all entire solutions to the equation \begin{equation*} f'f” - a(z)f^2 = b(z)e^{2c(z)}. \end{equation*} This completely solves two open questions posed by Gundersen and Yang in [Comput. Methods Funct. Theory 21 (2021), 605-617].

We study finite-horizon exit-time control of a one-dimensional affine diffusion, with an unbounded control acting on both drift and volatility. Since an admissible control can cancel the instantaneous volatility, the associated HJB equation is not uniformly parabolic. Moreover, the location of the degeneracy is not prescribed by the model coefficients but depends on the unknown value function and optimal feedback. Rather than relying on a viscosity-solution formulation, we work under suitable structural assumptions and prove that the value function is a $C^{1,2}$ classical solution throughout the interior of the state domain and is strictly convex in the state variable. Moreover, we construct a locally Lipschitz optimal feedback, explicitly identify the degenerate set as a curve, and show that the closed-loop drift on this curve equals the time derivative of its state coordinate. Finally, we prove invariance and inaccessibility before exit and derive exact squared-logarithmic asymptotics for approach probabilities.

This paper provides a comprehensive overview of the rate-distortion-perception framework, tracing its evolution from mathematical theory to practical deployment. We review the mathematical definition of the the mathematical definition of the Blau–Michaeli function and the associated coding theorems, examine computational methods for its evaluation, and discuss alternative formulations of the framework. Operational principles and design guidelines for modern image and video coding are also presented, highlighting the rate-distortion-perception framework as a rigorous foundation for developing next-generation perceptual compression systems.

The fast Fourier transform (FFT) and its real counterparts, the fast algorithms of the discrete cosine transform (DCT) and the discrete sine transform (DST), are frequently employed for the numerical evaluation of the Fourier cosine and Fourier sine transform, thereby acting as parametric quadrature rules based on finitely many samples of a given continuous function. But how accurate are the obtained results? In this paper we study the error occurring if the DCT of type I (DCT-I) and the DST of type I (DST-I) are applied to compute $\int_0^{\infty} f(x)\cos(2\pi xv)\d x$ and $\int_0^{\infty} f(x)\sin(2\pi xv)\d x$ for $f \in L^1(\Rp) \cap C(\Rp)$ and $v \in \Rp$, as well as the error occurring if the Chebyshev coefficients $\frac{2}{\pi}\int_0^{\pi} h(\cos\theta)\,\cos(n\theta)\d\theta$, $n \in \Np$, of a function $h \in C(I)$ on $I := [-1,1]$ are computed by the DCT-I. We present practicable, explicit error estimates under polynomial, exponential, and mixed decay conditions. For the Fourier cosine and sine transform, rational functions lead to a geometric decay in the frequency parameter $P$ combined with an algebraic plateau of order $L^{-3/2}$ in the sampling parameter $L$, while for meromorphic functions such as $\operatorname{sech}(\pi\,\cdot)$ the critical geometric rate is attained in both parameters. For Chebyshev approximation of a rational function with simple poles in $\C \setminus I$, the quadrature error has the critical geometric rate $\rho_0^{-P}$ with an explicit constant, where $\rho_0 > 1$ is the parameter of the largest Bernstein ellipse of analyticity. Several numerical examples illustrate the theory.

We give a geometric characterization of the pluricomplex Poisson kernel of bounded strongly linearly convex domains in $\mathbb C^{n+1}$ in terms of their foliation by all complex geodesic discs whose closure contains a fixed boundary point, thus confirming a conjecture posed by Bracci et al. in 2009, even in greater generality. The proof relies crucially on our previous results obtained in a series of two papers. We also revisit the homogeneous complex Monge–Ampère equation associated with the pluricomplex Poisson kernel and show that its solutions are far from unique in general by constructing a family of pairwise nonproportional continuous solutions on the unit ball in $\mathbb C^{n+1}$. This sharply contrasts with the uniqueness of solutions to the corresponding equation for the pluricomplex Green function.

The determinant approximation conjecture states that for a residually finite group $G$ and a chain $G_i$ of finite index normal subgroups with trivial intersection, the Fuglede-Kadison determinant of every matrix over the group algebra $\mathbb{Q} G$ is the limit of the Fuglede-Kadison determinants of the matrix reductions from $\mathbb{Q} G$ to $\mathbb{Q} (G/G_i)$. We disprove the conjecture by exhibiting an explicit element of the integral group ring of the Heisenberg group with determinant equal to $2$ and reduced determinants bounded by $\sqrt[3]{5}$.

We study the eigenvalues and eigenvectors of the adjacency matrix $A$ of the Erdős-Rényi graph $\mathbb G(N,p)$ in the regime $Np \gg (\log N)^2$. We establish optimal isotropic delocalization for the bulk eigenvectors $\boldsymbol u$, meaning that $\langle \boldsymbol v, \boldsymbol u\rangle^2 \leq \frac{C \log N}{N}$ with very high probability for any deterministic normalized $\boldsymbol v$. In addition, we prove local spectral universality in the bulk by showing that the local spectral statistics coincide with those of the GOE. The main tool of our proof is a local law for the resolvent of $A$, down to optimal spectral scales and with optimal error bounds. Its proof relies on a new approach to local laws that we call the Bernoulli flow. It is a characteristic flow method in which the usual Brownian process is replaced by a Bernoulli process, where each edge of the graph forms an independent Markov process that jumps at unit rate from closed to open. The spectral parameter flows according to a suitably constructed matrix-valued Bernoulli characteristic flow.

Estimation of the derivatives of a function with bounded high-order derivative in the presence of bounded measurement noise is considered, in a deterministic setting. Theoretical fundamental limitations of causal differentiators in terms of the lowest achievable worst-case differentiation error and desired properties such as exactness—the zero-error estimation of derivatives in the absence of noise—and robustness—the small sensitivity of the estimate under small perturbations—are established for the first time for arbitrary differentiation order. Differentiators that achieve these theoretically lowest bounds on their differentiation error are formally defined, fully characterized, and their properties are studied. In particular, deterministically optimal differentiators—those featuring optimal differentiation error bounds among the class of exact differentiators—are shown to exist by means of a novel construction exhibiting, in addition to exactness from the beginning, a very strong form of robustness.

We prove that triangles minimize the Monge-Ampère eigenvalue among bounded planar convex domains of a given area, thus resolving the two-dimensional case of Le's simplex conjecture for the Monge-Ampère eigenvalue (The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018)). The proof uses a neighbor-decreasing property of the Monge-Ampère eigenvalues of convex polygons along vertex elimination parallel chord movements.

We prove that Besicovitch's covering theorem holds for the parabolic metric $d_p((x_1,t_1),(x_2,t_2))=(|x_1-x_2|^{p}+|t_1-t_2|^{\frac{p}{2}})^{\frac{1}{p}}$ on $\mathbb{R}^n\times\mathbb{R}$ if and only if $p\geqslant 2$. If $p=2$, this answers a question posed by P. Mattila in the affirmative.

For every reproducing kernel Hilbert space $\mathcal{H}_K$ with bounded kernel $K$, we prove that the linear sampling widths $g_m^{\text{lin}}$ and Gelfand widths $c_n$ in the uniform norm satisfy $$ g_m^{\text{lin}}(B_{\mathcal H_K})_\infty \leq \frac{m+1}{m-n+1}\, c_n(B_{\mathcal H_K})_\infty\quad , \quad m\ge n. $$ In a certain sense, the leading factor $(m+1)/(m-n+1)$ on the right-hand side is sharp. We transfer the problem into the selection of a maximal volume subset of kernel translates (kernel $D$-optimal design). Apart from basic linear algebra, our proof employs low rank approximation techniques and a version of the Eckart-Young-Mirsky theorem. The same argument gives corresponding bounds for the selection of a reduced basis in Hilbert spaces.

We study packings of annuli of a common width, allowing each ring to nest inside the hole of a larger one. The objectives of maximizing contact area and cardinality diverge: area is superadditive in the radius, cardinality is not. Under superincreasing radii, every descending greedy maximizes every positive, strictly increasing, superadditive objective. More strongly, any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape in every dimension. This placement irrelevance holds unconditionally for at most three rings and fails at four in disks and squares; twin instances exclude every universal rule based only on the observable state. Write $\rho=\max_i(\sum_{j>i}r_j)/r_i$. The additive model has threshold exactly $1$. For disks we prove the exact global threshold $\tau=\varphi$, with no failure at $\rho\le\varphi$, for every finite inventory, even with independent hole radii. The key geometric theorem states that, under golden tail bounds, an entire disk list fits a circular container if and only if its three largest disks fit; this supplies the uniform exchange of parents that the threshold proof needs. The Tribonacci constant $T\approx1.83929$ remains the exact floor of a rigid subfamily. A dimension-reduction lemma transfers spherical sharpness results to all dimensions $d\ge2$, and a separate argument proves the golden threshold for at most five rings in those dimensions. For square pans, a Cartesian confinement criterion gives twins and a family proving $\tau_{\square}\le Y\approx1.6845$; its optimality is open. For independent holes, the exact universal area guarantee under $\rho\le\kappa<1$ is $\min(1,\kappa^{-2}-1)$, with threshold $1/\sqrt2$. The repository has 122 Lean theorems. Euclidean geometry, forest assembly and continuity remain written proofs; numerical checks do not substitute for them.

We discover an exponent-dependent phase transition phenomenon for optimal hypercontractivity: for every prescribed $q_0>2$, there exists a reversible continuous-time Markov chain on three state space with normalized spectral gap whose $(2,q)$-optimal hypercontractivity time satisfies $$ \text{$t_{\mathrm{opt}}(2,q)=\frac12\log(q-1)$ if and only if $q\ge q_0$},$$ whereas the strict inequality $t_{\mathrm{opt}}(2,q)>\frac12\log(q-1)$ holds for $2<q<q_0$.

We determine all possible signatures of flat Hermitian bundles over compact, connected, oriented surfaces of positive genus with nonempty boundary. If $\Sigma_{g,n}$ has genus $g\geq 1$ and $n\geq 1$ boundary components, then the signatures arising from representations $\pi_1(\Sigma_{g,n})\to\mathrm{U}(p,q)$ are exactly the integers $m$ satisfying \[ |m| \leq (p+q)(2g+n-2) -(2g-2)|p-q| -\min\{2,n|p-q|\}. \] Every such integer, including the two extremal values, is realized by a block-diagonal representation whose image is contained, after possibly interchanging $p$ and $q$, in $\mathrm{U}(1,1)^{\times\min\{p,q\}}\times\mathrm{U}(|p-q|)$.

We developed a mathematical model to describe the inflammatory dynamics during arthritis development and resolution, focusing on the interaction between three key immune cell populations within the joint tissue, namely neutrophils (N), sublining synovial tissue-resident macrophages (SL-TRM, T), and circulating monocytes that are recruited into the inflamed joint and differentiate into monocyte-derived macrophages (MoMac, M). An inflammatory signal (S), representing the response to an external arthritogenic stimulus, drives the recruitment and activation of neutrophils and MoMac, while SL-TRM exert a regulatory and protective role. We propose a minimal ordinary differential equation model of acute inflammatory response into a joint that couples an inflammatory signal S to the three cell populations (N, T, M) and incorporates recruitment, stimulation, nonlinear self-limitation and mutual regulation. We show that the equilibrium structure depends on three adimensional parameters A, B, C and derive a cubic equation characterizing nontrivial steady states. We perform an analytical study of equilibria and their stability, and identify a bifurcation at A = 1 separating a disease-free regime from a sustained inflammatory state. Numerical simulations illustrate typical transient dynamics under various scenarios and show the ability of the model to capture the transition from acute to chronic inflammation, as well as the resolution of inflammation under certain conditions. In conclusion, our model provides a framework to explore mechanisms underlying acute versus sustained joint inflammation and offers a basis for parameter estimation from experimental data, as well as future extensions toward evaluation of therapeutic strategies.

We consider a branching random walk on $\mathbb{Z}^2$ with critical offspring of mean $1$ and spatial motion governed by the jumps of a lazy simple random walk. For every site $x\in\mathbb{Z}^2$, we obtain a uniform asymptotical estimate for the local survival probability, i.e., the probability that there are particles at $x$ at large time $n$. Using Stein's method, we establish a Yaglom-type theorem for the number of particles at $x$ at time $n$ when $x$ is at distance of order $\sqrt{n}$ from the origin. Moreover, at the position occupied by a typical particle at time $n$, the number of particles at that site, normalized by $\log n$, converges in law to a Gamma distribution, thereby confirming a conjecture of Lalley and Zheng [Ann. Probab. 39 (2010), 327-368]. Finally, we prove that, conditional on local survival, the total number of particles at time $n$, divided by $n$, also converges weakly to a Gamma distribution.