Poisson convergence in the restricted k-partioning problem

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The randomized k-number partitioning problem is the task to distribute N i.i.d. random variables into k groups in such a way that the sums of the variables in each group are as similar as possible. The restricted k-partitioning problem refers to the case where the number of elements in each group is fixed to N/k. In the case k=2 it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case k>2 in the restricted problem and show that the vector of differences between the k sums converges to a k-1-dimensional Poisson point process.