The reconstructibility of finite abelian groups

Given a subset S of an abelian group G and an integer k greater than or equal to 1, the k-deck of S is the function that assigns to every T subset of or equal to G with at most k elements the number of elements g is an element of G with g + T subset of or equal to S. The reconstruction problem for an abelian group G asks for the minimal value of k such that every subset S of G is determined, up to translation, by its k-deck. This minimal value is the set-reconstruction number r(set)(G) of G; the corresponding value for multisets is the reconstruction number r(G). Previous work had given bounds for the set-reconstruction number of cyclic groups: Alon, Caro, Krasikov and Roditty [1] showed that r(set)(Z(n)) < log(2)n and Radcliffe and Scott [15] that r(set)(Z(n)) < 9(lnn)/(lnlnn). We give a precise evaluation of r(G) for all abelian groups G and deduce that r(set)(Z(n)) less than or equal to 6.