摘要 / Abstract 中文摘要 本文在动态欧氏几何 (Dynamic Euclidean Geometry, 见 zhang 2026a) 框架下审视 Kakeya 问题在 ℝ³ 中的 Hausdorff 维数。核心命题: 法向移动是维度升级的唯一操作途径。一维线段经两次法向移动分别升至二维与三维, 故合法 Kakeya 集在 ℝ³ 中 Hausdorff 维数 = 3 为该公理的直接推论。本文进一步指出, 若将 Perron tree 构造中未显式化的"面积可无限稀释"假设硬延至高维, 则构造中出现未经法向移动生成的假面积, 致有效维数低于 3 (作者直觉估 ≈2. 5), 该数值为假设施加的伪结果, 不在合法生成集范围内。Wang–Zahl (2023–2025) 以黏性公理为前提证得 ℝ³ 中 dimH = 3, 此黏性公理恰为否定 Perron tree 假设施设之条件, 与本框架一致。 English Abstract This paper examines the Hausdorff dimension of the Kakeya set in ℝ³ under the framework of Dynamic Euclidean Geometry (see zhang 2026a). The core proposition: normal translation is the unique operational pathway for dimensional upgrade. A 1D line segment undergoes two successive normal translations to reach 2D and 3D respectively; thus the Hausdorff dimension of a legitimate Kakeya set in ℝ³ equals 3 as a direct corollary of this axiom. The paper further notes that if the unstated assumption of "arbitrary dilutability of area" in the Perron tree construction is forcibly extended to higher dimensions, spurious area ungenerated by normal translation appears, yielding an effective dimension below 3 (author's intuitive estimate ≈2. 5) —a pseudo-result of the false assumption, outside the scope of legitimate generated sets. Wang–Zahl (2023–2025), proving dimH = 3 in ℝ³ under the stickiness axiom, precisely negate this false assumption; their result is consistent with the present framework.
zhigang zhang (Fri,) studied this question.