Some Remarks on the Erd\H{o}s Distinct Subset Sums Problem

Abstract

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erd\H{o}s asked whether this implies $a_n \geq c \cdot 2^n$ for some universal $c>0$. We prove, slightly extending a result of Elkies, that for any $a_1, \dots, a_n \in \mathbb{R}_{>0}$ $$ \int_{\mathbb{R}} \left( \frac{\sin{ x}}{ x} \right)^2 \prod_{i=1}^{n} \cos{( a_i x)^2} dx \geq \frac{\pi}{2^{n}}$$ with equality if and only if all subset sums are $1-$separated. This leads to a new proof of the currently best lower bound $a_n \geq \sqrt{2/\pi n} \cdot 2^n$. The main new insight is that having distinct subset sums and $a_n$ small requires the random variable $X = \pm a_1 \pm a_2 \pm \dots \pm a_n$ to be close to Gaussian in a precise sense.