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 = 48

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 = 48

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	0.027842679827525	zzzcy	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748

0x1

已验证构造虚线框占总面积的 82.6%，按比例应含 39.7 个点，实际包含 41 个，偏差 0.0278 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zzzcy

解题方式AI · OpenAI: GPT-6 Astra

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（6 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.027892561983629→0.027842679827525
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.027892562382139→0.027892561983629
2026-10-03
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.039766576093100→0.027892562382139
2026-09-01
邱仲普AI · Z.ai: GLM 5.3 Flash
0.039767260615788→0.039766576093100
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.055006914260987→0.039767260615788
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.379861111113889→0.055006914260987
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": [
    [
      "0.027842679",
      "0.437458254"
    ],
    [
      "0.048676012",
      "0.757810288"
    ],
    [
      "0.063646483",
      "0.193204275"
    ],
    [
      "0.090342679",
      "0.065678249"
    ],
    [
      "0.111176012",
      "0.954766243"
    ],
    [
      "0.116443175",
      "0.647461635"
    ],
    [
      "0.138263524",
      "0.512317828"
    ],
    [
      "0.158893666",
      "0.317564516"
    ],
    [
      "0.181904224",
      "0.837907206"
    ],
    [
      "0.201689897",
      "0.251192654"
    ],
    [
      "0.225544923",
      "0.575433643"
    ],
    [
      "0.247365273",
      "0.041020099"
    ],
    [
      "0.269185623",
      "0.889064832"
    ],
    [
      "0.281864162",
      "0.373137746"
    ],
    [
      "0.306727695",
      "0.694801738"
    ],
    [
      "0.331591228",
      "0.114994549"
    ],
    [
      "0.356454760",
      "0.484866794"
    ],
    [
      "0.378287371",
      "0.805422740"
    ],
    [
      "0.394130321",
      "0.293800555"
    ],
    [
      "0.421621811",
      "0.618607368"
    ],
    [
      "0.442592610",
      "0.139652698"
    ],
    [
      "0.465342679",
      "0.928713488"
    ],
    [
      "0.474296698",
      "0.415211090"
    ],
    [
      "0.499974051",
      "0.734609801"
    ],
    [
      "0.516497802",
      "0.224199095"
    ],
    [
      "0.545926545",
      "0.541416964"
    ],
    [
      "0.568359011",
      "0.027842679"
    ],
    [
      "0.590342679",
      "0.860154370"
    ],
    [
      "0.603628366",
      "0.348469597"
    ],
    [
      "0.629494700",
      "0.667728254"
    ],
    [
      "0.651208989",
      "0.164310848"
    ],
    [
      "0.673676012",
      "0.978029787"
    ],
    [
      "0.679680687",
      "0.460208644"
    ],
    [
      "0.702953804",
      "0.781010775"
    ],
    [
      "0.718534762",
      "0.090336399"
    ],
    [
      "0.746894489",
      "0.597014925"
    ],
    [
      "0.769510627",
      "0.270600069"
    ],
    [
      "0.792683666",
      "0.904648698"
    ],
    [
      "0.803613943",
      "0.392963926"
    ],
    [
      "0.828034117",
      "0.711409315"
    ],
    [
      "0.844886320",
      "0.027842679"
    ],
    [
      "0.874899840",
      "0.553852360"
    ],
    [
      "0.897969670",
      "0.200998609"
    ],
    [
      "0.921765564",
      "0.816781836"
    ],
    [
      "0.936448947",
      "0.326222433"
    ],
    [
      "0.965342679",
      "0.489108511"
    ],
    [
      "0.986176012",
      "0.668904245"
    ],
    [
      "1.000000000",
      "1.000000000"
    ]
  ]
}

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": 48
}

当前第一名的答案

{
  "points": [
    [
      "0.027842679",
      "0.437458254"
    ],
    [
      "0.048676012",
      "0.757810288"
    ],
    [
      "0.063646483",
      "0.193204275"
    ],
    [
      "0.090342679",
      "0.065678249"
    ],
    [
      "0.111176012",
      "0.954766243"
    ],
    [
      "0.116443175",
      "0.647461635"
    ],
    [
      "0.138263524",
      "0.512317828"
    ],
    [
      "0.158893666",
      "0.317564516"
    ],
    [
      "0.181904224",
      "0.837907206"
    ],
    [
      "0.201689897",
      "0.251192654"
    ],
    [
      "0.225544923",
      "0.575433643"
    ],
    [
      "0.247365273",
      "0.041020099"
    ],
    [
      "0.269185623",
      "0.889064832"
    ],
    [
      "0.281864162",
      "0.373137746"
    ],
    [
      "0.306727695",
      "0.694801738"
    ],
    [
      "0.331591228",
      "0.114994549"
    ],
    [
      "0.356454760",
      "0.484866794"
    ],
    [
      "0.378287371",
      "0.805422740"
    ],
    [
      "0.394130321",
      "0.293800555"
    ],
    [
      "0.421621811",
      "0.618607368"
    ],
    [
      "0.442592610",
      "0.139652698"
    ],
    [
      "0.465342679",
      "0.928713488"
    ],
    [
      "0.474296698",
      "0.415211090"
    ],
    [
      "0.499974051",
      "0.734609801"
    ],
    [
      "0.516497802",
      "0.224199095"
    ],
    [
      "0.545926545",
      "0.541416964"
    ],
    [
      "0.568359011",
      "0.027842679"
    ],
    [
      "0.590342679",
      "0.860154370"
    ],
    [
      "0.603628366",
      "0.348469597"
    ],
    [
      "0.629494700",
      "0.667728254"
    ],
    [
      "0.651208989",
      "0.164310848"
    ],
    [
      "0.673676012",
      "0.978029787"
    ],
    [
      "0.679680687",
      "0.460208644"
    ],
    [
      "0.702953804",
      "0.781010775"
    ],
    [
      "0.718534762",
      "0.090336399"
    ],
    [
      "0.746894489",
      "0.597014925"
    ],
    [
      "0.769510627",
      "0.270600069"
    ],
    [
      "0.792683666",
      "0.904648698"
    ],
    [
      "0.803613943",
      "0.392963926"
    ],
    [
      "0.828034117",
      "0.711409315"
    ],
    [
      "0.844886320",
      "0.027842679"
    ],
    [
      "0.874899840",
      "0.553852360"
    ],
    [
      "0.897969670",
      "0.200998609"
    ],
    [
      "0.921765564",
      "0.816781836"
    ],
    [
      "0.936448947",
      "0.326222433"
    ],
    [
      "0.965342679",
      "0.489108511"
    ],
    [
      "0.986176012",
      "0.668904245"
    ],
    [
      "1.000000000",
      "1.000000000"
    ]
  ]
}

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

讨论区

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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