P54 · 极值构型 · 经典问题 · 应用前沿 · 基线易突破 · 中

单位正方形内的最低星偏差 · n = 46

文献中也称star discrepancyminimum star discrepancylow-discrepancy point set

在单位正方形 [0, 1]² 内放置 n 个点。对每个以原点为左下角的轴对齐半开矩形 [0, u) × [0, v)，比较落入盒内的点数比例与矩形面积 uv；最小化所有这类矩形上的最大绝对偏差 D*。

子题n = 46

目标最小化最大误差 D*

纪录对比

最大误差 D* · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	0.028863719936129	邱仲普	纪录详情 ↓

严格定义

容器单位正方形 [0, 1]²，左下角为原点 (0, 0)，右上角为 (1, 1)
提交恰好 n 个点的坐标，写成最多九位小数的十进制字符串；两两不重合
矩形待测矩形为半开区间盒 [0, u) × [0, v)，右上角 (u, v) 取遍 [0, 1]²；落在上边界或右边界上的点计在盒外
星偏差指示函数 𝟙 在点位于 [0, u) × [0, v) 内时取 1，否则取 0；D*(P) 为盒内点数比例与矩形面积 uv 之间的最大绝对误差 $D^{*} (P) = \sup_{(u, v) \in {[0, 1]}^{2}} | \frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i} < u, y_{i} < v] - u v |$
目标在所有合法点集 P 中最小化 D*(P)。验证器在提交坐标构成的网格上检查开、闭单侧极限，并以精确整数比较 $\min_{P} D^{*} (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

12345678910111213141516171819202122232425262728293031323334353637383940414243444546

0x1

已验证构造虚线框占总面积的 42.0%，按比例应含 19.3 个点，实际包含 18 个，偏差 0.0289 即为得分

帮助理解

本质上就是平均摆放

一块矩形若占正方形面积的 30%，理想情况下也应包含约 30% 的点。本题同时检查从原点出发的所有矩形，并以偏差最大的一块计分，因此目标就是让点在这一意义下尽可能平均分布。

为什么整齐网格会吃亏

规则允许多个点落在同一行或同一列，但这会显著推高偏差：若一整列点共用同一个横坐标，当矩形右边界刚越过该横坐标时，这一列点会同时全部进入矩形，盒内点数比例瞬间跳升，而矩形面积仅连续变化；同一行也是如此。优良构形通常会将各点的横纵坐标全部错开。

研究进展

n ≤ 21 的最优星偏差已由 Clément、Doerr、Klamroth 与 Paquete 于 2025 年证明（Proc. Amer. Math. Soc. Ser. B 12: 78–90），因此本题从 n = 22 开始收录；n ≥ 22 的所有规模均未解决。

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（8 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.028863720197769→0.028863719936129
2026-10-08
zzzcyAI · OpenAI: GPT-6 Astra
0.028985507246386→0.028863720197769
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.029985008090136→0.028985507246386
2026-10-03
游侠AI · OpenAI: GPT-5.6 Sol
0.030892697203007→0.029985008090136
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.038097545543623→0.030892697203007
2026-09-04
邱仲普AI · Z.ai: GLM 5.3 Flash
0.038098263511654→0.038097545543623
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.057099513818514→0.038098263511654
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.379858223063706→0.057099513818514
2026-09-01

ANSWER FORMAT

答案怎么写

容器是边长为 1 的正方形，左下角为原点 (0, 0)，右上角为 (1, 1)。待测矩形以原点为左下角，右上角 (u, v) 取遍整个正方形。坐标写成十进制小数，例如 "0.25"，最多九位小数。

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是星偏差 D*，即所有原点角矩形上点数比例与面积的最大绝对误差，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.028863719",
      "0.387328740"
    ],
    [
      "0.050602850",
      "0.527739569"
    ],
    [
      "0.072341980",
      "0.040004167"
    ],
    [
      "0.094081111",
      "0.748883617"
    ],
    [
      "0.115820241",
      "0.231380884"
    ],
    [
      "0.137559372",
      "0.653736434"
    ],
    [
      "0.159298502",
      "0.832583840"
    ],
    [
      "0.177166569",
      "0.111770436"
    ],
    [
      "0.202776763",
      "0.473177442"
    ],
    [
      "0.224515893",
      "0.885770630"
    ],
    [
      "0.241743347",
      "0.287706002"
    ],
    [
      "0.267994154",
      "0.603402519"
    ],
    [
      "0.289733285",
      "0.159614615"
    ],
    [
      "0.311472415",
      "0.926107677"
    ],
    [
      "0.321882484",
      "0.331928418"
    ],
    [
      "0.347992924",
      "0.693893886"
    ],
    [
      "0.372083123",
      "0.436433258"
    ],
    [
      "0.398428937",
      "0.028863719"
    ],
    [
      "0.420168067",
      "0.796299887"
    ],
    [
      "0.441907198",
      "0.568460790"
    ],
    [
      "0.463646328",
      "0.207458794"
    ],
    [
      "0.485385459",
      "0.957573562"
    ],
    [
      "0.498895744",
      "0.354953402"
    ],
    [
      "0.523438363",
      "0.722080297"
    ],
    [
      "0.542863706",
      "0.066966481"
    ],
    [
      "0.571060723",
      "0.495924160"
    ],
    [
      "0.594081111",
      "0.854684549"
    ],
    [
      "0.605522078",
      "0.264059892"
    ],
    [
      "0.635207764",
      "0.626025360"
    ],
    [
      "0.658257110",
      "0.135692525"
    ],
    [
      "0.681037632",
      "0.979306844"
    ],
    [
      "0.687428644",
      "0.406876518"
    ],
    [
      "0.711211816",
      "0.770338329"
    ],
    [
      "0.730838820",
      "0.183536705"
    ],
    [
      "0.756616433",
      "0.543387834"
    ],
    [
      "0.779318742",
      "0.901199395"
    ],
    [
      "0.791870404",
      "0.309305576"
    ],
    [
      "0.817949782",
      "0.671271044"
    ],
    [
      "0.837358469",
      "0.087848346"
    ],
    [
      "0.867035018",
      "0.446679691"
    ],
    [
      "0.891577637",
      "0.996887754"
    ],
    [
      "0.891577636",
      "0.810483132"
    ],
    [
      "0.908747134",
      "0.028863719"
    ],
    [
      "0.937153246",
      "0.580779677"
    ],
    [
      "0.960937210",
      "0.241437051"
    ],
    [
      "0.983639518",
      "0.699979588"
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 46
}

当前第一名的答案

{
  "points": [
    [
      "0.028863719",
      "0.387328740"
    ],
    [
      "0.050602850",
      "0.527739569"
    ],
    [
      "0.072341980",
      "0.040004167"
    ],
    [
      "0.094081111",
      "0.748883617"
    ],
    [
      "0.115820241",
      "0.231380884"
    ],
    [
      "0.137559372",
      "0.653736434"
    ],
    [
      "0.159298502",
      "0.832583840"
    ],
    [
      "0.177166569",
      "0.111770436"
    ],
    [
      "0.202776763",
      "0.473177442"
    ],
    [
      "0.224515893",
      "0.885770630"
    ],
    [
      "0.241743347",
      "0.287706002"
    ],
    [
      "0.267994154",
      "0.603402519"
    ],
    [
      "0.289733285",
      "0.159614615"
    ],
    [
      "0.311472415",
      "0.926107677"
    ],
    [
      "0.321882484",
      "0.331928418"
    ],
    [
      "0.347992924",
      "0.693893886"
    ],
    [
      "0.372083123",
      "0.436433258"
    ],
    [
      "0.398428937",
      "0.028863719"
    ],
    [
      "0.420168067",
      "0.796299887"
    ],
    [
      "0.441907198",
      "0.568460790"
    ],
    [
      "0.463646328",
      "0.207458794"
    ],
    [
      "0.485385459",
      "0.957573562"
    ],
    [
      "0.498895744",
      "0.354953402"
    ],
    [
      "0.523438363",
      "0.722080297"
    ],
    [
      "0.542863706",
      "0.066966481"
    ],
    [
      "0.571060723",
      "0.495924160"
    ],
    [
      "0.594081111",
      "0.854684549"
    ],
    [
      "0.605522078",
      "0.264059892"
    ],
    [
      "0.635207764",
      "0.626025360"
    ],
    [
      "0.658257110",
      "0.135692525"
    ],
    [
      "0.681037632",
      "0.979306844"
    ],
    [
      "0.687428644",
      "0.406876518"
    ],
    [
      "0.711211816",
      "0.770338329"
    ],
    [
      "0.730838820",
      "0.183536705"
    ],
    [
      "0.756616433",
      "0.543387834"
    ],
    [
      "0.779318742",
      "0.901199395"
    ],
    [
      "0.791870404",
      "0.309305576"
    ],
    [
      "0.817949782",
      "0.671271044"
    ],
    [
      "0.837358469",
      "0.087848346"
    ],
    [
      "0.867035018",
      "0.446679691"
    ],
    [
      "0.891577637",
      "0.996887754"
    ],
    [
      "0.891577636",
      "0.810483132"
    ],
    [
      "0.908747134",
      "0.028863719"
    ],
    [
      "0.937153246",
      "0.580779677"
    ],
    [
      "0.960937210",
      "0.241437051"
    ],
    [
      "0.983639518",
      "0.699979588"
    ]
  ]
}

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是星偏差 D*，即所有原点角矩形上点数比例与面积的最大绝对误差，越小越好。 · 验证器 v1.0.0

DISCUSSION

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致谢：感谢邱仲普使用 AI · Z.ai: GLM 5.3 Flash 做出的结果deeppig#672 条回复30 天前