Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2410.06612

Some observations on Erdős matrices

Raghavendra Tripathi
View original source ↗Read source document ↗Export metadata ↓

Abstract

In a seminal paper in 1959, Marcus and Ree proved that every $n\times n$ bistochastic matrix $A$ satisfies $\|A\|_{\operatorname{F}}^2\leq \max_{σ\in S_n}A_{i,σ(i)}$ where $S_n$ is the symmetric group on $\{1, \ldots, n\}$. Erdős asked to characterize the bistochastic matrices for which the equality holds in the Marcus--Ree inequality. We refer to such matrices as Erdős matrices. While this problem is trivial in dimension $n=2$, the case of dimension $n=3$ was only resolved recently in~\cite{bouthat2024question} in 2023. We prove that for every $n$, there are only finitely many $n\times n$ Erdős matrices. We also give a characterization of Erdős matrices that yields an algorithm to generate all Erdős matrices in any given dimension. We also prove that Erdős matrices can have only rational entries. This answers a question of~\cite{bouthat2024question}.

Explore related subjects

math.MGmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Raghavendra Tripathi. 2024-12-13. Some observations on Erdős matrices. https://arxiv.org/abs/2410.06612

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗
arXiv2026

Finite and Dynamic Stability Horizons for Nearest-Neighbor Future Structures

Nearest-neighbor graphs are discrete objects whose membership may change under small perturbations of the underlying coordinates. We establish an explicit stability guarantee for finite labeled configurations in an arbitrary metric space. If the gap between the kth and (k+1)st distances is positive, simultaneous per-label perturbations smaller than one quarter of that gap preserve the directed k-nearest-neighbor membership. The factor four is sharp under the stated uniform displacement assumptions, and every fixed deterministic construction based solely on the labeled neighbor family is consequently invariant. Under an interval-valid Lipschitz bound for labeled information-space trajectories, the same result yields a certified lower bound on the first possible rewiring time, with a refinement for label-specific motion bounds. We apply the finite theorem to frozen standardized Taylor-Green future-information coordinates [d_B, log A_B] for 585 particles at three observed time strata. Outward-rounded interval arithmetic certified a sufficient perturbation radius of approximately 1.023 x 10^-6, and a separately implemented checker within the same research workflow verified the rank and distance-margin calculations. An outcome-blind audit then examined whether the construction supported a numerical continuous-time horizon. Because the complete information map involved discrete clustering and boundary reconstruction and lacked an analytic derivative bound, validated dense-time bound, or certified modulus of continuity, the application was correctly classified as DISCRETE_ONLY with STOP_NO_INTERVAL_VALID_BOUND. Thus finite local structural invariance is proved and numerically certified for the frozen configuration, while temporal specialization and predictive generalization remain separate questions requiring additional evidence.

math.MG↗
arXiv2026

Equivalence of Landscape and Erosion Distances for Persistence Diagrams

This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.

math.MG↗