A Generalisation on Erdős Distinct Subset Sums Problem

This paper investigates the Erdős distinct subset sums problem in ${\mathbb{Z}}^k$. Beyond the classical variance method, using alternative statistical quantities like $\mathbb{E}[\mathrm{\parallel }X{\mathrm{\parallel }}_1]$ and $\mathbb{E}[\mathrm{\parallel }X{\mathrm{\parallel }}_3^3]$ can yield better bounds in certain dimensions. This innovation improves previous low-dimensional results and provides a framework for choosing suitable methods depending on the dimension.