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We derive Maximal Kakeya estimates for functions over$\mathbb{Z}/N\mathbb{Z}$ proving the Maximal Kakeya conjecture for$\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright[HW18]. The proof involves using polynomial method and linear algebratechniques from [Dha21, Ars21a, DD21] and generalizing a probabilistic methodargument from [DD22]. As another application we give lower bounds for the sizeof $(m,\epsilon)$-Kakeya sets over $\mathbb{Z}/N\mathbb{Z}$. Using these ideaswe also give a new, simpler, and direct proof for Maximal Kakeya bounds overfinite fields (which were first proven in [EOT10]) with almost sharp constants.

Maximal and $(m,\epsilon)$-Kakeya bounds over $\mathbb{Z}/N\mathbb{Z}$ for general $N$