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Let $A$ be a set of $n$ integers uniformly distributed in $\{0,\dots,N-1\}$. Let $S$ be the set of subset-sums modulo $N$ of $A$. Let $f_{n,N}(k)$ be the probability that $|S|=k$.

Is there an expression for $f_{n,N}(k)$? In particular, I am interested in the case where $N=2^n$.

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You might be interested in this paper:

Li, Jiyou; Wan, Daqing, Counting subset sums of finite Abelian groups, J. Comb. Theory, Ser. A 119, No. 1, 170-182 (2012). ZBL1229.05289.

Abstract:

In this paper, we obtain an explicit formula for the number of zero-sum $k$-element subsets in any finite abelian group.

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  • $\begingroup$ Thank you so much for improving this answer! $\endgroup$
    – Joe Franklin
    Commented May 30, 2022 at 1:04

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