  In this paper, we present a uniqueness result for solutions to the
Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption
of an a priori spacetime bound. We show that this a priori bound is satisfied
for factorized solutions to the hierarchy which come from solutions of the
nonlinear Schr\"{o}dinger equation. In this way, we obtain a periodic analogue
of the uniqueness result on $\mathbb{R}^3$ previously proved by Klainerman and
Machedon, except that, in the periodic setting, we need to assume additional
regularity. In particular, we need to work in the Sobolev class $H^{\alpha}$
for $\alpha>1$. By constructing a specific counterexample, we show that, on
$\mathbb{T}^3$, the existing techniques don't apply in the endpoint case
$\alpha=1$. This is in contrast to the known results in the non-periodic
setting, where the these techniques are known to hold for all $\alpha \geq 1$.
  In our analysis, we give a detailed study of the crucial spacetime estimate
associated to the free evolution operator. In this step of the proof, our
methods rely on lattice point counting techniques based on the concept of the
determinant of a lattice. This method allows us to obtain improved bounds on
the number of lattice points which lie in the intersection of a plane and a set
of radius $R$, depending on the number-theoretic properties of the normal
vector to the plane. We are hence able to obtain a sharp range of admissible
Sobolev exponents for which the spacetime estimate holds.
