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Abstract for Stan Wagon and Herbert S. Wilf, When are subset sums 
equidistributed modulo m?

For a triple $(n,t,m)$ of positive integers, we attach to each $t$-subset $S=\{a_1,\ldots ,a_t\}\subseteq \{1,\ldots ,n\}$ the sum $f(S)=a_1+\cdots +a_t$ (modulo $m$). We ask: for which triples $(n,t,m)$ are the ${n\choose t}$ values of $f(S)$ uniformly distributed in the residue classes mod $m$? The obvious necessary condition, that $m$ divides ${n\choose t}$, is not sufficient, but a $q$-analogue of that condition is both necessary and sufficient, namely:
$${{q^m-1}\over {q-1}}\quad {\rm divides\ the\ Gaussian\ polynomial}\quad \gauss{n}{t}{q}.$$
We show that this condition is equivalent to: for each divisor $d>1$ of $m$, we have $t\ {\rm mod}\, d>n\ {\rm mod}\, d$. Two proofs are given, one by generating functions and another via a bijection. We study the analogous question on the full power set of $[n]$: given $(n,m)$; when are the $2^n$ subset sums modulo $m$ equidistributed into the residue classes? Finally we obtain some asymptotic information about the distribution when it is not uniform, and discuss some open questions.
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