Gaussian Lower Bounds for Distinct Subset Sums in Higher Dimensions

For sequences of $n$ nonnegative integer vectors in $\mathbb Z^k$ with distinct subset sums, we prove, for fixed $k$, the uniform lower bound $M\ge (C_k-o(1))2^{n/k}/\sqrt n$ for the largest coordinate, where $C_k=\sqrt{\frac{2k}π}\, \frac{(k+1)^{(k-1)/(2k)}}{\sqrt{\lfloor (k+1)^2/4\rfloor}}.$ The proof compares a Gaussian-kernel expectation with a sum over a translated lattice, uniformly over the vector sequence. We optimise the resulting quadratic-form constant using a vertex second-moment certificate. We show that $C_k$ also exceeds the variance constant obtained by optimising an ellipsoidal geometry.