Erdős Problem #362

Let \(A\subseteq \mathbb{N}\) be a finite set of size \(N\). Is it true that, for any fixed \(t\), there are \[\ll \frac{2^N}{N^{3/2}}\] many \(S\subseteq A\) such that \(\sum_{n\in S}n=t\)? Erdős and Moser [Er65] proved the first bound with an additional factor of \((\log n)^{3/2}\). This was removed by Sárközy and Szemerédi [SaSz65], thereby answering the first question in the affirmative. Stanley [St80] has shown that this quantity is maximised when \(A=\{-\lfloor \frac{N-1}{2}\rfloor,\ldots,\lfloor\frac{N}{2}\rfloor\}\).