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Let $A=\{1,...,n\}$. Two subsets of $A$, not necessarily distinct, chosen uniformly at random. What is the probability that both subsets have the same sum? Alternatively, is there a known upper bound?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...

This is inspired by the recent question How many solutions $\pm1\pm2\pm3…\pm n=0$. The oeis entries A063865 linked to this question and A292476/A156700 for the related one "How many solutions $\pm1\...

Edit: This is a question related to my other post, stated in a much more concrete way I think. I am interested in anything (ideas, references) related to the following problem: Suppose that $A \...