A Generalisation on Erdős Distinct Subset Sums Problem

TL;DR

This work addresses the Erdős distinct subset sums problem extended to $\mathbb{Z}^k$ by bounding the maximum coordinate $M$ of an $M$-bounded distinct subset sum sequence. It introduces a Statistical Bridge framework based on a random variable $X = \sum_{i=1}^n \varepsilon_i a_i$ with $\varepsilon_i\in\{\pm\tfrac12\}$, and derives three moment-based bounds by analyzing $\mathbb{E}[\|X\|_1]$ and $\mathbb{E}[\|X\|_3^3]$ in conjunction with $p$-norm volume identities. The main contributions are two explicit asymptotic lower bounds on $M$—a first-moment bound for $k\le 4$ and a third-moment bound for $4<k\le 6$—and a dimensional-optimality perspective that situates these against Costa et al.’s variance-based bound for larger $k$. The results provide a dimension-aware toolkit for approaching Erdős’s problem in higher dimensions and open avenues for using higher-order statistics to tighten bounds across different dimensional regimes.

Abstract

This paper investigates the Erdős distinct subset sums problem in $\mathbb{Z}^k$. Beyond the classical variance method, using alternative statistical quantities like $\mathbb{E}[\|X\|_1]$ and $\mathbb{E}[\|X\|_3^3]$ 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.

Figures (1)

• Figure 1: Comparison of coefficients relative to $k$ for three distinct methods

Theorems & Definitions (12)

• Definition 1.1
• Theorem 1.2
• Definition 3.1
• Proposition 3.2
• Proposition 3.3
• Lemma 3.4
• proof
• Theorem 4.1
• proof
• Theorem 4.2