cahlen/hausdorff-dimension-spectrum

---
license: cc-by-4.0
task_categories:
- tabular-classification
tags:
- mathematics
- continued-fractions
- fractal-geometry
- hausdorff-dimension
- spectral-theory
- gpu-computation
size_categories:
- 1M<n<10M
---

# Hausdorff Dimension Spectrum: All Subsets of {1,...,20}

First complete computation of the Hausdorff dimension of E_A for all 2^20 - 1 = 1,048,575 non-empty subsets A of {1,...,20}. Computed via transfer operator + Chebyshev collocation (N=40) on NVIDIA RTX 5090.

This dataset does not exist anywhere in the published literature.

## Files

- `spectrum_n5.csv` — All 31 subsets of {1,...,5}
- `spectrum_n10.csv` — All 1,023 subsets of {1,...,10}
- `spectrum_n20.csv` — All 1,048,575 subsets of {1,...,20} (main dataset, 61 MB)
- `metadata_*.json` — Computation parameters for each run
- `run_n20.log` / `run_n20_recompute.log` — Computation logs

## CSV Schema

| Column | Description |
|--------|-------------|
| `subset_mask` | Bitmask encoding the subset |
| `subset_digits` | Human-readable digit set, e.g. `{1,2,3}` |
| `cardinality` | Number of digits in subset |
| `max_digit_in_subset` | Largest digit |
| `dimension` | Hausdorff dimension dim_H(E_A), 15 significant digits |

## Key Values

| Subset | dim_H(E_A) |
|--------|-----------|
| {1,2} | 0.5313 |
| {1,2,3,4,5} | 0.8368 (Zaremba semigroup) |
| {1,...,10} | 0.9540 |
| {1,...,20} | 0.9653 |

## Method

Transfer operator with Chebyshev collocation at N=40 nodes. Hausdorff dimension computed as the unique s where the leading eigenvalue equals 1. Bisection with 55 steps for ~15 digit precision. Power iteration with 300 iterations per eigenvalue.

## Hardware

NVIDIA RTX 5090 (32 GB). Full n=20 computation: 4,343 seconds.

## Citation

```bibtex
@dataset{humphreys2026hausdorff,
author = {Humphreys, Cahlen},
title = {Hausdorff Dimension Spectrum for Continued Fraction Cantor Sets},
year = {2026},
publisher = {Hugging Face},
url = {https://huggingface.co/datasets/cahlen/hausdorff-dimension-spectrum}
}
```

## Source

- **Code**: [hausdorff-spectrum](https://github.com/cahlen/idontknow/tree/main/scripts/experiments/hausdorff-spectrum)
- **Findings**: [Digit 1 Dominance](https://bigcompute.science/findings/hausdorff-digit-one-dominance/)
- **Project**: [bigcompute.science](https://bigcompute.science)
- **MCP Server**: `mcp.bigcompute.science` (22 tools, no auth)
- **AGENTS.md**: [Contribution guide](https://github.com/cahlen/idontknow/blob/main/AGENTS.md)

## Citation

```bibtex
@misc{humphreys2026hausdorff_dimension_spectrum,
author = {Humphreys, Cahlen and Claude (Anthropic)},
title = {Hausdorff Dimension Spectrum for Continued Fraction Cantor Sets},
year = {2026},
publisher = {Hugging Face},
url = {https://huggingface.co/datasets/cahlen/hausdorff-dimension-spectrum}
}
```

Human-AI collaborative work (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification. CC BY 4.0.

license: 知识共享署名4.0（CC BY 4.0）协议
task_categories:
- 表格分类（tabular-classification）
tags:
- 数学（mathematics）
- 连分式（continued-fractions）
- 分形几何（fractal-geometry）
- 豪斯多夫维数（hausdorff-dimension）
- 谱理论（spectral-theory）
- GPU计算（gpu-computation）
size_categories:
- 100万 < 样本量 < 1000万

# 豪斯多夫维数谱：集合{1,…,20}的所有子集

首次完成了对集合{1,…,20}的全部2^20 - 1 = 1,048,575个非空子集A对应的E_A的豪斯多夫维数的计算。本次计算通过转移算子（transfer operator）+切比雪夫配点法（Chebyshev collocation，N=40）在NVIDIA RTX 5090上完成。本数据集尚未见于已发表的学术文献中。

## 文件列表

- `spectrum_n5.csv` — 集合{1,…,5}的全部31个子集数据
- `spectrum_n10.csv` — 集合{1,…,10}的全部1,023个子集数据
- `spectrum_n20.csv` — 集合{1,…,20}的全部1,048,575个子集数据（主数据集，大小61 MB）
- `metadata_*.json` — 各次运行的计算参数文件
- `run_n20.log` / `run_n20_recompute.log` — 计算日志文件

## CSV数据格式

| 列名 | 含义说明 |
|--------|-------------|
| `subset_mask` | 编码子集的位掩码（bitmask） |
| `subset_digits` | 人类可读形式的数字集合，例如`{1,2,3}` |
| `cardinality` | 子集的元素基数（即元素个数） |
| `max_digit_in_subset` | 子集中包含的最大数字 |
| `dimension` | 豪斯多夫维数dim_H(E_A)，保留15位有效数字 |

## 关键示例值

| 子集 | dim_H(E_A) |
|--------|-----------|
| {1,2} | 0.5313 |
| {1,2,3,4,5} | 0.8368（扎雷巴半群） |
| {1,…,10} | 0.9540 |
| {1,…,20} | 0.9653 |

## 计算方法

采用N=40个节点的切比雪夫配点法结合转移算子。豪斯多夫维数通过求解使系统主导特征值等于1的唯一实数s得到。计算过程中使用二分法迭代55次以实现约15位数字的精度，每次特征值求解采用300次幂迭代（power iteration）。

## 计算硬件

NVIDIA RTX 5090（32 GB显存）。完整的n=20规模计算耗时4,343秒。

## 引用格式

bibtex
@dataset{humphreys2026hausdorff,
author = {Humphreys, Cahlen},
title = {Hausdorff Dimension Spectrum for Continued Fraction Cantor Sets},
year = {2026},
publisher = {Hugging Face},
url = {https://huggingface.co/datasets/cahlen/hausdorff-dimension-spectrum}
}

bibtex
@misc{humphreys2026hausdorff_dimension_spectrum,
author = {Humphreys, Cahlen and Claude (Anthropic)},
title = {Hausdorff Dimension Spectrum for Continued Fraction Cantor Sets},
year = {2026},
publisher = {Hugging Face},
url = {https://huggingface.co/datasets/cahlen/hausdorff-dimension-spectrum}
}

## 来源

- **代码仓库**：[hausdorff-spectrum](https://github.com/cahlen/idontknow/tree/main/scripts/experiments/hausdorff-spectrum)
- **研究发现**：[Digit 1 Dominance](https://bigcompute.science/findings/hausdorff-digit-one-dominance/)
- **项目主页**：[bigcompute.science](https://bigcompute.science)
- **MCP服务器**：`mcp.bigcompute.science`（包含22个工具，无需认证）
- **贡献指南**：[AGENTS.md](https://github.com/cahlen/idontknow/blob/main/AGENTS.md)

本工作为人类与AI智能体（AI Agent）协作成果（Cahlen Humphreys + Claude），尚未经过独立同行评审。所有代码与数据均开源以供验证。采用CC BY 4.0协议。