Some Remarks on the Erdős Distinct Subset Sums Problem

Let $\{a_1,\dots ,a_n\}\subset \mathbb{N}$ 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. Erdős asked whether this implies $a_n\ge c\cdot 2^n$ for some universal $c>0$. We prove, slightly extending a result of Elkies, that for any $a_1,\dots ,a_n\in {\mathbb{R}}_{>0}$ $${\int }_{\mathbb{R}}{\left(\frac{\sin x}{x}\right)}^2\prod _{i=1}^n\cos (a_{i}x{)}^{2}dx\ge \frac{\pi }{2^n}$$ with equality if and only if all subset sums are $1-$separated. This leads to a new proof of the currently best lower bound $a_n\ge \sqrt{2\mathrm{/}\pi n}\cdot 2^n$. The main new insight is that having distinct subset sums and $a_n$ small requires the random variable $X=\pm a_1\pm a_2\pm \cdots \pm a_n$ to be close to Gaussian in a precise sense.