A Fraction Dimensional Kakeya Compression

Abstract

Recently, it was shown in arXiv:2607.27629 that under a very general condition, a generic family of curved Kakeya sets in ${\mathbb{R}}^3$ not trivially contained in a hypersurface has Hausdorff dimension strictly greater than 2. In this paper we leverage large language models to give an explicit construction of a family of curves failing the worst compression phenomenon in arXiv:2607.27629, but whose union has Hausdorff dimension and upper box dimension $5/2$. The example shows that there are Kakeya compression phenomena besides the trivial example of concentrating near a hypersurface. This answers a question raised in section 1.9 of arXiv:2607.27629. The models used are ChatGPT 5.6 Sol and ChatGPT 6 Astra.