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In this paper we study the restriction estimate for the flat disk over finitefields. Mockenhaupt and Tao initially studied this problem but their resultswere addressed only for dimensions $n=4,6$. We improve and extend their resultsto all dimensions $n\geq 6$. More precisely, we obtain the sharp $L^2 \to L^r$estimates, which cannot be proven by applying the usual Stein-Tomas argumentover a finite field even with the optimal Fourier decay estimate on the flatdisk. One of main ingredients is to discover and analyze an explicit form ofthe Fourier transform of the surface measure on the flat disk. In addition,based on the recent results on the restriction estimates for the paraboloids,we address improved restriction estimates for the flat disk beyond the $L^2$restriction estimates.

Restriction estimates for the flat disks over finite fields