Reconstructing infinite sets of integers

[摘要] For a set of integers A subset of or equal to Z and k greater than or equal to 1 the k-deck of A is the function d(A,k) defined on sets S of k integers by d(A,k)(S) = \{i is an element of Z \ {s + i \ s is an element of S} subset of or equal to A}\. Our main result is that for k greater than or equal to 3, a set for which the k-deck only takes finite values is determined up to translation by its k-deck and one finite non-zero value of its (k -1)-deck. This generalizes a result of Radcliffe and Scott (Electron. J. Combin. 6 (1999), R20) which proved a weaker form of this statement for k = 3. In order to establish this result, we generalize Kelly's Lemma for finite graphs to infinite sets of integers. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-08-01

[关键词] reconstruction;deck;Kelly's Lemma