Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.21417 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918355611418624 |
|---|---|
| author | Cobeli, Cristian Nguyen, The Zaharescu, Alexandru |
| author_facet | Cobeli, Cristian Nguyen, The Zaharescu, Alexandru |
| contents | Let $\mathcal{M}$ be a set with $M$ elements, let $ψ:\mathcal{M}\to\mathcal{M}$ be a bijective involution, and let~$\boldsymbol{\mathcal{X}}_ψ$ be the set of sequences $(x_1,\dots,x_M)\in\mathcal{M}^M$ with the property that $x_{M+1-j} = ψ(x_j)$ for $1\le j\le M$. This framework can be used to infer the possible distribution of sequences, such as the modular ones, that pose challenges for conventional methods.
We prove that when $M$ is even, there exists a limit probability density function that weighs the parameter $k$ that counts the appearances of the elements of $\mathcal{M}$ among the terms of sequences $\textbf{x}\in\boldsymbol{\mathcal{X}}_ψ$. It turns out that the number of fixed points of $ψ$ influences the probability density function, which decomposes into two pieces, each multiplied by complementary factors, and the smaller of the two pieces appears only when $k$ is even.
Applying the model, we find a threshold from which almost all sequences contain related terms with prescribed frequencies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21417 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A statistical model for points expanding in higher dimensions while being tied to bijective involutions Cobeli, Cristian Nguyen, The Zaharescu, Alexandru Number Theory Probability primary 11N69, secondary 49J55, 65C10 Let $\mathcal{M}$ be a set with $M$ elements, let $ψ:\mathcal{M}\to\mathcal{M}$ be a bijective involution, and let~$\boldsymbol{\mathcal{X}}_ψ$ be the set of sequences $(x_1,\dots,x_M)\in\mathcal{M}^M$ with the property that $x_{M+1-j} = ψ(x_j)$ for $1\le j\le M$. This framework can be used to infer the possible distribution of sequences, such as the modular ones, that pose challenges for conventional methods. We prove that when $M$ is even, there exists a limit probability density function that weighs the parameter $k$ that counts the appearances of the elements of $\mathcal{M}$ among the terms of sequences $\textbf{x}\in\boldsymbol{\mathcal{X}}_ψ$. It turns out that the number of fixed points of $ψ$ influences the probability density function, which decomposes into two pieces, each multiplied by complementary factors, and the smaller of the two pieces appears only when $k$ is even. Applying the model, we find a threshold from which almost all sequences contain related terms with prescribed frequencies. |
| title | A statistical model for points expanding in higher dimensions while being tied to bijective involutions |
| topic | Number Theory Probability primary 11N69, secondary 49J55, 65C10 |
| url | https://arxiv.org/abs/2602.21417 |