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In this paper, we present a uniqueness result for solutions to theGross-Pitaevskii hierarchy on the three-dimensional torus, under the assumptionof an a priori spacetime bound. We show that this a priori bound is satisfiedfor factorized solutions to the hierarchy which come from solutions of thenonlinear Schr\"{o}dinger equation. In this way, we obtain a periodic analogueof the uniqueness result on $\mathbb{R}^3$ previously proved by Klainerman andMachedon, except that, in the periodic setting, we need to assume additionalregularity. 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-periodicsetting, 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 estimateassociated to the free evolution operator. In this step of the proof, ourmethods rely on lattice point counting techniques based on the concept of thedeterminant of a lattice. This method allows us to obtain improved bounds onthe number of lattice points which lie in the intersection of a plane and a setof radius $R$, depending on the number-theoretic properties of the normalvector to the plane. We are hence able to obtain a sharp range of admissibleSobolev exponents for which the spacetime estimate holds.

On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy