A note on Erdős matrices and Marcus\unicode{x2013}Ree inequality
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| Format: | Preprint |
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2025
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| _version_ | 1866911065579716608 |
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| author | Kushwaha, Aman Tripathi, Raghavendra |
| author_facet | Kushwaha, Aman Tripathi, Raghavendra |
| contents | In 1959, Marcus and Ree proved that any bistochastic matrix $A$ satisfies $Δ_n(A):= \max_{σ\in S_n}\sum_{i=1}^{n}A(i, σ(i))-\sum_{i, j=1}^n A(i, j)^2 \geq 0$. Erdős asked to characterize the bistochastic matrices satisfying $Δ_n(A)=0$. This problem remains largely open, and very recently, a complete list of such matrices was obtained in dimension $n=3$ by Bouthat, Mashreghi, and Morneau-Guérin. Soon after, Tripathi proved that there were only finitely many such matrices in any dimension $n$. In this paper, we continue the investigation initiated in these two works. We characterize all $4\times 4$ bistochastic matrices satisfying $Δ_4(A)=0$. Furthermore, we show that for $n\geq 3$, $Δ_n(A)=α$ has uncountably many solutions when $α\in (0, (n-1)/4)$. This answers a question raised in [Tripathi, R., $\textit{Some observations on Erdős matrices}$, Linear Algebra and Its Applications 708 (2025)]. We also extend the Marcus\unicode{x2013}Ree inequality to infinite bistochastic arrays and bistochastic kernels. Our investigation into $4\times 4$ Erdős matrices also leads to several intriguing questions of independent interest. We propose several questions and conjectures and present numerical evidence for them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09542 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on Erdős matrices and Marcus\unicode{x2013}Ree inequality Kushwaha, Aman Tripathi, Raghavendra Metric Geometry Functional Analysis 15B99, 15B51, 15A45 (Primary) 15A80 (Secondary) In 1959, Marcus and Ree proved that any bistochastic matrix $A$ satisfies $Δ_n(A):= \max_{σ\in S_n}\sum_{i=1}^{n}A(i, σ(i))-\sum_{i, j=1}^n A(i, j)^2 \geq 0$. Erdős asked to characterize the bistochastic matrices satisfying $Δ_n(A)=0$. This problem remains largely open, and very recently, a complete list of such matrices was obtained in dimension $n=3$ by Bouthat, Mashreghi, and Morneau-Guérin. Soon after, Tripathi proved that there were only finitely many such matrices in any dimension $n$. In this paper, we continue the investigation initiated in these two works. We characterize all $4\times 4$ bistochastic matrices satisfying $Δ_4(A)=0$. Furthermore, we show that for $n\geq 3$, $Δ_n(A)=α$ has uncountably many solutions when $α\in (0, (n-1)/4)$. This answers a question raised in [Tripathi, R., $\textit{Some observations on Erdős matrices}$, Linear Algebra and Its Applications 708 (2025)]. We also extend the Marcus\unicode{x2013}Ree inequality to infinite bistochastic arrays and bistochastic kernels. Our investigation into $4\times 4$ Erdős matrices also leads to several intriguing questions of independent interest. We propose several questions and conjectures and present numerical evidence for them. |
| title | A note on Erdős matrices and Marcus\unicode{x2013}Ree inequality |
| topic | Metric Geometry Functional Analysis 15B99, 15B51, 15A45 (Primary) 15A80 (Secondary) |
| url | https://arxiv.org/abs/2503.09542 |