arXiv subjects
Publications and source records attributed to Pengyu Yang.
We consider the eigenform locus $\mathcal{E}_D$ in $\mathcal{H}(1,1)$ where $D$ is not a square. We prove that for any translation surface $(X,ω) \in \mathcal{E}_D$, the set of non-uniquely ergodic directions has Hausdorff dimension $1/2$, except when $D=5$ and $(X,ω)$ lies in the Teichmüller curve generated by the regular decagon.
We study Birkhoff genericity for a simple uniformly expanding diagonal flow on $\mathrm{SL}_{n+1}(\mathbb R)/\mathrm{SL}_{n+1}(\mathbb Z)$, with initial points restricted to affine subspaces of the expanding horospherical orbit through the identity coset. We prove that almost every point on such an affine subspace is Birkhoff generic, except possibly in two situations: either the defining matrix of the affine subspace has Diophantine exponent at least $n$, or the affine subspace is arbitrarily well approximable by affine subspaces of dimension $(r-1)$ defined over a real number field of degree $m\ge 2$, with $n+1=mr$. As applications, we obtain Dirichlet non-improvability and logarithmic density results for almost every point on these affine subspaces.
We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central ingredient is a measure classification result of Einsiedler and Lindenstrauss for actions of rank two diagonalizable groups on quotients of products of $\mathrm{SL}_2$. This rigidity result together with an application of the Siegel mass formula reduces the density problem to a counting problem on a certain affine variety. We solve that counting problem using the determinant method of Bombieri-Pila and Heath-Brown.
Constructing large-scale datasets for the GitHub issue resolution task is crucial for both training and evaluating the software engineering capabilities of Large Language Models (LLMs). However, the existing GitHub issue resolution data construction pipeline is challenging and labor-intensive. We identify three key limitations in existing pipelines: (1) test patches collected often omit binary file changes; (2) the manual construction of evaluation environments is labor-intensive; and (3) the fail2pass validation phase requires manual inspection of test logs and writing custom parsing code to extract test status from logs. In this paper, we propose SWE-Factory, a fully automated issue resolution data construction pipeline, to resolve these limitations. First, our pipeline automatically recovers missing binary test files and ensures the correctness of test patches. Second, we introduce SWE-Builder, a LLM-based multi-agent system that automates evaluation environment construction. Third, we introduce a standardized, exit-code-based log parsing method to automatically extract test status, enabling a fully automated fail2pass validation. Experiments on 671 real-world GitHub issues across four programming languages show that our method can effectively construct valid evaluation environments for GitHub issues at a reasonable cost. For example, with GPT-4.1 mini, our SWE-Builder constructs 337 valid task instances out of 671 issues, at $0.047 per instance. Our ablation study further shows the effectiveness of different components of SWE-Builder. We also demonstrate through manual inspection that our exit-code-based fail2pass validation method is highly accurate, achieving an F1 score of 0.99. Additionally, we conduct an exploratory experiment to investigate whether we can use SWE-Factory to enhance models' software engineering ability.
We prove a local-global principle for representations of binary by quaternary quadratic forms. One of the main ingredients is a recent measure rigidity result of Einsiedler and Lindenstrauss for diagonalizable actions on quotients of products of $\mathrm{SL}_2$'s. Based on this, it suffices to show that limits of the uniform measures on the associated rank one adelic toral packets have more entropy than one half of the maximal entropy. The latter is proved using the Siegel mass formula and the determinant method as developed by Bombieri and Pila as well as Heath-Brown.
The creation of large-scale, high-fidelity quantum computers is not only a fundamental scientific endeavour in itself, but also provides increasingly robust proofs of quantum computational advantage (QCA) in the presence of unavoidable noise and the dynamic competition with classical algorithm improvements. To overcome the biggest challenge of photon-based QCA experiments, photon loss, we report new Gaussian boson sampling (GBS) experiments with 1024 high-efficiency squeezed states injected into a hybrid spatial-temporal encoded, 8176-mode, programmable photonic quantum processor, Jiuzhang 4.0, which produces up to 3050 photon detection events. Our experimental results outperform all classical spoofing algorithms, particularly the matrix product state (MPS) method, which was recently proposed to utilise photon loss to reduce the classical simulation complexity of GBS. Using the state-of-the-art MPS algorithm on the most powerful supercomputer EI Capitan, it would take > $10^{42}$ years to construct the required tensor network for simulation, while our Jiuzhang 4.0 quantum computer takes 25.6 $μ$s to produce a sample. This work establishes a new frontier of QCA and paves the way to fault-tolerant photonic quantum computing hardware.
We prove a version of Linnik's basic lemma uniformly over the base field using theta-series and geometric invariant theory in the spirit of Khayutin's approach (Duke Math. J., 168(12), 2019). As an application, we establish entropy bounds for limits of invariant measures on homogeneous toral sets in GL(4) of biquadratic, cyclic, or dihedral type.
To address the issue of excessive quantum resource requirements in Kuperberg's algorithm for the dihedral hidden subgroup problem, this paper proposes a distributed algorithm based on the function decomposition. By splitting the original function into multiple subfunctions and distributing them to multiple quantum nodes for parallel processing, the algorithm significantly reduces the quantum circuit depth and qubit requirements for individual nodes. Theoretical analysis shows that when $n\gg t$ ($t$ is the number of quantum nodes), the time complexity of the distributed version is optimized from $2^{O(\sqrt{n})}$ (the traditional algorithm's complexity) to $2^{o(\sqrt{n-t})}$. Furthermore, we carried out the simulation on the Qiskit platform, and the accuracy of the algorithm is verified. Compared to the original algorithm, the distributed version not only reduces the influence of circuit depth and noise, but also improves the probability of measurement success.
For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real-analytic submanifold under a diagonal flow. This extends the earlier result of Shah in the case of non-degenerate submanifolds. We apply the above dynamical result to show that if the affine span of a real-analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet-improvable.
In Diophantine approximation, the notion of singular vectors was introduced by Khintchine in the 1920's. We study the set of singular vectors on an affine subspace of $\mathbb{R}^n$. We give an upper bound of its Hausdorff dimension in terms of the Diophantine exponent of the parameter of the affine subspace.
On the space $\mathcal{L}_{n+1}$ of unimodular lattices in $\mathbb{R}^{n+1}$, we consider the standard action of $a(t)=\mathrm{diag}(t^n,t^{-1},\ldots,t^{-1})\in \mathrm{SL}(n+1,\mathbb{R})$ for $t>1$. Let $M$ be a nondegenerate submanifold of an expanding horospherical leaf in $\mathcal{L}_{n+1}$. We prove that for all $x\in M\setminus E$ and $t>1$, if $μ_{x,t}$ denotes the normalized Lebesgue measure on the ball of radius $t^{-1}$ around $x$ in $M$, then the translated measure $a(t)μ_{x,t}$ get equidistributed $\mathcal{L}_{n+1}$ as $t\to\infty$, where $E$ is a union of countably many lower dimensional submanifolds of $M$. In particular, if $μ$ is an absolutely continuous probability measure on $M$, then $a(t)μ$ gets equidistributed in $\mathcal{L}_{n+1}$ as $t\to\infty$. This result implies the non-improvability of Dirichlet's Diophantine approximation theorem for almost every point on a $C^{n+1}$-submanifold of $\mathbb{R}^n$ satisfying a non-degeneracy condition, answering a question arising from the work of Davenport and Schmidt (1969).
We show that under the action of $\mathrm{diag}(e^{nt},e^{-r_1(t)},\ldots,e^{-r_n(t)})\in\mathrm{SL}(n+1,\mathbb{R})$, where $r_i(t)\to\infty$, on the space of unimodular lattices in $\mathbb{R}^{n+1}$, the translates of any fixed-sized piece of a `non-degenerate' smooth curve, or a shrinking piece of size $e^{-t}$ about almost any point of the curve, get equidistributed in the space as $t\to\infty$. From this, it follows that the weighted Dirichlet approximation theorem cannot be improved for almost all points on any non-degenerate $C^{2n}$ curve in $\mathbb{R}^n$. This result extends the corresponding result for analytic curves due to Shah (2009) and answers some questions inspired by the work of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008).
Let $X=\text{SL}_3(\mathbb{R})/\text{SL}_3(\mathbb{Z})$, and $g_t=\text{diag}(e^{2t}, e^{-t}, e^{-t})$. Let $ν$ denote the push-forward of the normalized Lebesgue measure on a segment of a straight line in the expanding horosphere of $\{g_t\}_{t>0}$, under the map $h\mapsto h\text{SL}_3(\mathbb{Z})$ from $\text{SL}_3(\mathbb{R})$ to $X$. We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families of measures on $X$: (1) $g_t$-translates of $ν$ as $t\to\infty$. (2) averages of $g_t$-translates of $ν$ over $t\in[0,T]$ as $T\to\infty$. (3) $g_{t_i}$-translates of $ν$ for some $t_i\to\infty$. We apply this dynamical result to show that Lebesgue-almost every point on the planar line $y=ax+b$ is not Dirichlet-improvable if and only if $(a,b)\notin\mathbb{Q}^2$.
Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of $Δ(H)\backslash L^n$ with respect to a balanced line bundle, where $Δ(H)$ is the image of $H$ diagonally embedded in $L^n$.
We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/Γ$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.
In large scale distributed linear transform problems, coded computation plays an important role to effectively deal with "stragglers" (distributed computations that may get delayed due to few slow or faulty processors). We propose a coded computation strategy, referred to as diagonal code, that achieves the optimum recovery threshold and the optimum computation load. This is the first code that simultaneously achieves two-fold optimality in coded distributed linear transforms. Furthermore, by leveraging the idea from random proposal graph theory, we design two random codes that can guarantee optimum recovery threshold with high probability but with much less computation load. These codes provide order-wise improvement over the state-of-the-art. Moreover, the experimental results show significant improvement compared to both uncoded and existing coding schemes.