Zi-Jie Gu

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 \[ C_k=\sqrt{\frac{2k}{\pi}}\, \frac{(k+1)^{(k-1)/(2k)}}{\sqrt{\lfloor (k+1)^2/4\rfloor}}. \] The proof compare...