2024

Liu, Jiayin

This thesis studies the Hausdorff dimension of variants of Kakeya sets in Rn. It consists of an introduction and three papers. In paper [A], we define Kakeya sets in the n-th Heisenberg group and show the sharp lower bound is 3 for the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group. In papers [B] and [C], we define circular (s, t)-Furstenberg sets F in R2. We prove that the Hausdorff dimension dimH(F) ⩾ t 3 + s if 0 < s ⩽ 1 and 0 < t ⩽ 3 and dimH(F) ⩾ (2s−1)t+s if 1/2 < s ⩽ 1 and 0 < t ⩽ 1 in [B]. Moreover, we show the sharp lower bound dimH(F) ⩾ s + t if 0 < t ⩽ s ⩽ 1 in [C].