Reconstructing Subsets of Zn

In this paper we consider the problem of reconstructing a subset A ⊂ Z n , up to translation, from the collection of its subsets of size k, given up to translation (its k-deck). Results of Alon, Caro, Krasikov, and Roditty [1] show that this is possible provided k > log 2 n. Mnukhin [10] showed that every subset of Z n of size k is reconstructible from its (k − 1)-deck, provided k ≥ 4. We show that when n is prime every subset of Z n is reconstructible from its 3-deck; that for arbitrary n almost all subsets of Z n are reconstructible from their 3-decks; and that for any n every subset of Z n is reconstructible from its 9α(n)-deck, where α(n) is the number of distinct prime factors of n. We also comment on analogous questions for arbitrary groups, in particular the cube Z n 2. Our approach is to generalize the problem to that of reconstructing arbitrary rational functions on Z n. We prove — by analysing the interaction between the ideal structure of the group ring Q Z n and the operation of pointwise multiplication of functions — that with a suitable definition of deck every rational-valued function on Z n is reconstructible from its 9α(n)-deck.