Curly Kakeya: The Hausdorff dimension of sets containing circles or line segments in many directions

Abstract

In this paper, we establish a lower bound for the Hausdorff dimension of a set 𝐾 ⊂ ℝ 𝑛 that contains a dilated and translated copy of every meridian, that is, of every ( 𝑑 − 2 ) -dimensional sphere passing through the poles of 𝑆 𝑑 − 1 , a ( 𝑑 − 1 ) -dimensional sphere, with 3 ≤ 𝑑 ≤ 𝑛 , and 𝑆 𝑑 − 1 contained in ℝ 𝑛 . Under this assumption, we have d i m 𝐻 ⁡ ( 𝐾 ) ≥ 𝑑 − 1 . This result is closely related to, and reminiscent of, the classical Kakeya set problem. We build upon this connection, employing similar techniques to show that if 𝐾 ⊂ ℝ 𝑛 contains a unit straight line segment in every direction corresponding to a smooth curve on 𝑆 𝑛 − 1 , then its Hausdorff dimension is ≥2.