arXiv subjects

Pál Zsámboki

Publications and source records attributed to Pál Zsámboki.

11 recordsLinked to original sources

arXiv2026

Improved bounds for the double cap conjecture

In 1974, Witsenhausen asked for the maximum possible density $α_n$ of a measurable subset $A$ of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ such that $A$ contains no pair of orthogonal vectors. For $n=3$, the best known lower bound is $1 - 1/\sqrt{2} = 0.29289\dots$, obtained from the natural "double cap" construction of two opposite spherical caps, which is conjectured to be optimal for all $n$ by Gil Kalai. In this paper, we use a novel approach to establish an upper bound of $α_3\le 0.2953$, improving the previous best known bound $0.2977$ due to Bekker et al. (2025). Our approach combines harmonic-analytic arguments with the geometric fractional chromatic number of finite graphs, recently introduced by Ambrus et al. (2024). In this framework, any finite subset of the sphere yields an upper bound for $α_n$, and we obtain our bound by identifying an appropriate 33-element point set through a large-scale computer search. The same method can also be used in higher dimensions to yield potential improvements of the best known bounds.

math.CO↗

arXiv2026

Post-Norm can Resharpen Attention

Length Generalization is the essential capacity of autonomous agents to perform tasks in longer contexts than those encountered during training. To systematically study this feat, we test how well models can approximate the next token distributions in algorithmic tasks. This is to take into account the realistic possibility of multiple next tokens being legal. We present a prototypical benchmark for this line of study: in the Set Complement Task, the model needs to output a uniform distribution over tokens not in the input. We prove a theorem that states simple transformers can length generalize on this task, however, with performance degradation due to attention dispersion. A mechanistic reading of how dispersion takes effect lets us discover a remedy: Post-Norm can Resharpen Attention. We present experimental evidence to support this idea. We also show that Exponential Moving Averages can help the issue of noisy gradients that arises when many next tokens are legal. We validate the general applicability of our proposed methods on a suite of formal language experiments. Our source code will be available upon publication.

cs.LG↗

arXiv2026

Commutative classifying space for simplicial groups

In this paper, we introduce a simplicial analog of classifying spaces for commutativity which classify principal bundles with commutativity structure on their transition functions. Our construction $\overline W(τ,K)$, which takes as input a simplicial group $K$ and a cosimplicial group $τ$ that encodes the additional structure such as commutativity, is a variation of the $\overline W$-construction for simplicial groups. Our main result shows that the geometric realization of our $\overline W(τ,K)$ is homotopy equivalent to the topological classifying space $B(τ,|K|)$.

math.AT↗

arXiv2025

Diverse beam search to find densest-known planar unit distance graphs

This paper addresses the problem of determining the maximum number of edges in a unit distance graph (UDG) of $n$ vertices using computer search. An unsolved problem of Paul Erdős asks the maximum number of edges $u(n)$ a UDG of $n$ vertices can have. Those UDGs that attain $u(n)$ are called "maximally dense." In this paper, we seek to demonstrate a computer algorithm to generate dense UDGs for vertex counts up to at least 100. Via beam search with an added visitation metric, our algorithm finds all known maximally dense UDGs up to isomorphism at the push of a button. In addition, for $15 < n$, where $u(n)$ is unknown, i) the algorithm finds all previously published densest UDGs up to isomorphism for $15 < n \le 30$, and ii) the rate of growth of $u(n)/n$ remains similar for $30 < n$. The code and database of over 60 million UDGs found by our algorithm can be found at https://codeberg.org/zsamboki/dbs-udg.

math.CO↗

arXiv2025

The fractional chromatic number of the plane is at least 4

We prove that the fractional chromatic number $χ_f(\mathbb R^2)$ of the unit distance graph of the Euclidean plane is greater than or equal to $4$. Interestingly, however, we cannot present a finite subgraph $G$ of the plane such that $χ_f(G)\ge 4$. Instead, we utilize the concept of the geometric fractional chromatic number $χ_{gf}(G)$, which was introduced recently in connection with density bounds for 1-avoiding sets. First, as $G$ ranges over finite subgraphs of the plane, we establish that the supremum of $χ_f(G)$ is the same as that of $χ_{gf}(G)$. The proof exploits the amenability of the group of Euclidean transformations in dimension 2 and, as such, we do not know whether the analogous statement holds in higher dimensions. We then present a specific planar unit distance graph $G$ on 27 vertices such that $χ_{gf}(G)=4$, and conclude $χ_f(\mathbb R^2)\ge 4$ as a corollary. As another main result we show that the finitary fractional chromatic number and the Hall ratio of the plane are equal. As a consequence, we conclude that there exist finite unit distance graphs with independence ratio $\frac{1}{4}+\varepsilon$, while we conjecture that the value $\frac{1}{4}$ cannot be reached.

math.CO↗

arXiv2025

Do Attention Heads Compete or Cooperate during Counting?

We present an in-depth mechanistic interpretability analysis of training small transformers on an elementary task, counting, which is a crucial deductive step in many algorithms. In particular, we investigate the collaboration/competition among the attention heads: we ask whether the attention heads behave as a pseudo-ensemble, all solving the same subtask, or they perform different subtasks, meaning that they can only solve the original task in conjunction. Our work presents evidence that on the semantics of the counting task, attention heads behave as a pseudo-ensemble, but their outputs need to be aggregated in a non-uniform manner in order to create an encoding that conforms to the syntax. Our source code will be available upon publication.

cs.LG↗

arXiv2024

Training BERT Models to Carry Over a Coding System Developed on One Corpus to Another

This paper describes how we train BERT models to carry over a coding system developed on the paragraphs of a Hungarian literary journal to another. The aim of the coding system is to track trends in the perception of literary translation around the political transformation in 1989 in Hungary. To evaluate not only task performance but also the consistence of the annotation, moreover, to get better predictions from an ensemble, we use 10-fold crossvalidation. Extensive hyperparameter tuning is used to obtain the best possible results and fair comparisons. To handle label imbalance, we use loss functions and metrics robust to it. Evaluation of the effect of domain shift is carried out by sampling a test set from the target domain. We establish the sample size by estimating the bootstrapped confidence interval via simulations. This way, we show that our models can carry over one annotation system to the target domain. Comparisons are drawn to provide insights such as learning multilabel correlations and confidence penalty improve resistance to domain shift, and domain adaptation on OCR-ed text on another domain improves performance almost to the same extent as that on the corpus under study. See our code at https://codeberg.org/zsamboki/bert-annotator-ensemble.

cs.CL↗

arXiv2023

The density of planar sets avoiding unit distances

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set avoiding unit distances cannot exceed $1/4$. Our argument implies the upper bound of $0.2470$.

math.MG↗

arXiv2022

A homotopical Skolem--Noether theorem

The classical Skolem--Noether Theorem [Giraud, 71] shows us (1) how we can assign to an Azumaya algebra $A$ on a scheme $X$ a cohomological Brauer class in $H^2(X,\mathbf G_m)$ and (2) how Azumaya algebras correspond to twisted vector bundles. The Derived Skolem--Noether Theorem [Lieblich, 09] generalizes this result to weak algebras in the derived 1-category locally quasi-isomorphic to derived endomorphism algebras of perfect complexes. We show that in general for a co-family $\mathscr C^\otimes$ of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving in particular the above correspondences. For a totally supported perfect complex $E$ over a quasi-compact and quasi-separated scheme $X$, the long exact sequence on homotopy group sheaves splits giving equalities $π_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Perf}}} E,\mathrm{id}_E)=π_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Alg}}\mathop{\mathrm{Perf}}}\mathop{\mathbf R\mathrm{End} E},\mathrm{id}_{\mathop{\mathbf R\mathrm{End} E}})$ for $i\ge1$. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and ind-coherent sheaves and crystals in Derived Algeraic Geometry in characteristic 0.

math.AG↗

arXiv2021

Twisted forms of perfect complexes and Hilbert 90

Automorphisms of a perfect complex naturally have the structure of an $\infty$-group: the 1-morphisms are quasi-isomorphisms, the 2-morphisms are homotopies, etc. This article starts by proving some basic properties of this $\infty$-group. We go on to study the deformation theory of this stack of $\infty$-groups and give a criterion for this stack to be formally smooth. The classifying stack of this $\infty$-group classifies forms of a complex. We discuss a version of Hilbert 90 for perfect complexes.

math.AG↗

arXiv2018

On the $\infty$-stack of complexes over a scheme

We study fppf descent for enhanced derived categories. We revisit the work of [HS] and [TV08] in a lax context. More precisely, we construct a Cartesian and coCartesian fibration ${}^{\mathrm{op}}\mathscr D^+_S\rightarrow N(\mathop{\mathrm{Sch}}_S)$ whose fibre over an $S$-scheme $T$ is the opposite $\mathscr D^+(T)^{\mathrm{op}}$ of the quasicategory of bounded below complexes of $\mathscr O_T$-modules. We show that this fibration satisfies fppf-descent for schemes.

math.AG↗