We estimate the
norm of the restriction to a totally geodesic submanifold of the
eigenfunctions of the Laplace–Beltrami operator on the standard flat torus
,
. We
reduce getting correct bounds to counting lattice points in the intersection of some
-transverse
bands on the sphere. Moreover, we prove the correct bounds for
rational totally geodesic submanifolds of arbitrary codimension.
In particular, we verify the conjecture of Bourgain–Rudnick on
-restriction estimates for
rational hyperplanes. On ,
we prove the uniform
restriction bounds for closed geodesics. On
, we obtain
explicit
restriction estimates for the totally geodesic submanifolds, which improve the
corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge.
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