TITLE: probability of non-existence of a sum subset
QUESTION [5 upvotes]: Given a list of distinct integers $a_1,\dots,a_n$ chosen uniformly from the interval $(-2^n,2^n)$, what is the probability that the list contains $\underline{NO\mbox{ }SUBSETS}$ that sum to another integer $b:|b|<2^n$? (what is the probability that the np complete problem SUBSET SUM does not have a subset sum)
Is the odds that there is no subset that sums to b (when b is fixed) is $\frac{1}{2}\pm\epsilon$ or $\epsilon$ or $1-\epsilon$?

REPLY [2 votes]: The probability that a particular subset of size $k$ has sum $b$ is bounded by $\frac{C}{2^n \sqrt{k}}$, (maximized by $b=0$). Thus the probability that some subset has sum $b$ is at most the sum over $k\leq n$, which is roughly $C/\sqrt{n}$.
[EDIT:] The above is for numbers chosen uniformly with repetition. However, this makes little difference, as the probability of a repetition is much smaller, of order $n^2 2^{-n}$.