Reconstructing Subsets of Zn

In this paper we consider the problem of reconstructing a subsetA⊂Zn, up to translation, from the collection of its subsets of sizek, given up to translation (itsk-deck). Results of Alon, Caro, Krasikov, and Roditty (1989,J. Combin. Theory Ser. B47, 153–161) show that this is possible providedk>log2 n. Mnukhin (1992,Acta. Appl. Math.29, 83–117) showed that every subset of Znof sizekis reconstructible from its (k−1)-deck, providedk⩾4. We show that whennis prime every subset of Znis reconstructible from its 3-deck; that for arbitrarynalmost all subsets of Znn are reconstructible from their 3-decks; and that for anynevery subset of Znis reconstructible from its 9α(n)-deck, whereα(n) is the number of distinct prime factors ofn. We also comment on analogous questions for arbitrary groups, in particular the cube Zn2. Our approach is to generalize the problem to that of reconstructing arbitrary rational functions on Zn. We prove—by analysing the interaction between the ideal structure of the group ring QZnand the operation of pointwise multiplication of functions—that with a suitable definition of deck every rational-valued function on Znis reconstructible from its 9α(n)-deck.