Additive Decompositions, Random Allocations, and Threshold Phenomena

An additive decomposition of a set I of nonnegative integers is an expression of I as the arithmetic sum of two other such sets. If the smaller of these has p elements, we have a p-decomposition. If I is obtained by randomly removing nα integers from {0, ..., n − 1}, decomposability translates into a balls-and-urns problem which we start to investigate (for large n) by first showing that the number of p-decompositions exhibits a threshold phenomenon as α crosses a p-dependent critical value. We then study in detail the distribution of the number of 2-decompositions. For this last case we show that the threshold is sharp and we establish the threshold function. 1 Definitions, Notations and basic Properties. For integers i and sets of integers J and K, we use the fairly standard notations i+ J = J + i = {i+ j : j ∈ J} , J +K = {j + k : (j, k) ∈ J ×K} . In this paper, we restrict ourselves mostly to sets of nonnegative integers, though we do allow ourselves the first notation with negative i, so long as −i is less than min J. Definition 1.1 A set I of nonnegative integers is said to be decomposable if there exist sets J,K, each with at least two elements, such that I = J +K. We also say that I is J-decomposable, that (J,K) is a decomposition of I, and that J is a decomposition factor for I. Thus, a set of two integers is never decomposable; a set of three integers is decomposable iff it is of the form {0, a, 2a} ; and a set {a, b, c, d} of four integers a < b < c < d is decomposable iff a+d = b+c. (In this paper, we consider the decomposability of finite sets only.) Issues of decomposability of integer sets arise occasionally in mathematics and computer science, most immediately in the context of the factorization of polynomials, since plainly a necessary condition for a polynomial in one indeterminate with coefficients in some field to split into two factors is that its set of exponents be decomposable. The more restrictive notion of irredundent decomposition, where it is required that the map (j, k) 7→ j + k be one-to-one, gives a necessary and sufficient condition for a factorization to exist within the class of Minkowski polynomials, i.e., polynomials in R [X] with 0-1 coefficients. Decompositions are, of course, generally not unique; further, if (J,K) is a decomposition of I, then min I = minJ +minK, and since any 0 ≤ i ≤ min I can be written as j + k with 0 ≤ j ≤ minJ and 0 ≤ k ≤ minK, resulting in (I − i) = (J − j) + (K − k), decomposability is seen to be, in effect, a property of equivalence classes of subsets of N under translation. Equally, if I = J +K then also I = (J −min J) + (K +minJ). Therefore: 1 Lemma 1.2 If I is decomposable, then it admits a decomposition factor containing 0. So in investigating whether a set J may be a factor of decomposition of I, we need, and shall, only consider sets containing 0. This remark lies at the root of the critical phenomena to be investigated. Also note that for one of the factors to be a subset of I, it is necessary and sufficient that the other factor contain 0 ( I = J+K withK ⊂ I implies min I = minJ+minK ≥ min J+min I, so min J ≤ 0); and for both factors to be subsets of I, it is necessary and sufficient that I contain 0. Lemma 1.3 A necessary and sufficient condition for a set J containing 0 to be a decomposition factor of I is: for any i in I, there is some (not necessarily unique) j in J such that i− j + J ⊂ I. Proof If I = J + K, then for any i ∈ I there is a j ∈ J such that i − j ∈ K, implying i − j + J ⊂ K + J ⊂ I. Conversely, if the condition is true, let K = {k ∈ N : k + J ⊂ I} . By construction J +K ⊂ I, and by hypothesis I ⊂ J +K. Note that for an arbitrary J , the condition in Lemma 1.3 is necessary and sufficient for J−minJ to be a decomposition factor of I; J itself is one, iff the condition holds with the added requirement that j ≤ i. Of course, for J containing 0, the definition of K might as well read K = {k ∈ I : k + J ⊂ I} . For a given J , we now characterize the sets I which are not J-decomposable. This follows by negating the condition in Lemma 1.3: Proposition 1.4 Let ΦJ be the set of fixed-point free maps from J to itself, and for φ ∈ ΦJ set: Aφ = {I ⊂ N : (∃i ∈ I) (∀j ∈ J) (i− j + φ (j) / ∈ I)} . Then {I ⊂ N : J is not a decomposition factor for I} = ⋃ φ∈ΦJ Aφ. (We exclude maps φ with fixed points simply because they all give empty Aφ’s.) To simplify the study of the problem of decomposition as defined above, a main restriction is to consider the decomposition for a fixed size of the decomposition factor J . Definition 1.5 A set I of nonnegative integers is said to be p-decomposable if there exist sets J,K, one of which, say J , has exactly p elements, such that I = J +K. In this paper we will focus on the p-decomposability of a set I of integers. In the next section we define a natural probabilistic model and we show that there exists a threshold phenomenon for p-decomposability as a function of an appropriate control parameter. In the following sections we investigate 2-decomposability more specifically and we give a detailed study, providing asymptotic distributions dependent on the control parameter, and proving in particular the existence of a sharp threshold for which we establish the threshold function. 2 Random model and critical behavior for p-decomposability. We show in this section that for a given integer p ≥ 2, the p-decomposability of random sets of integers according to an appropriate probability distribution exhibits a critical behavior similar to that observed in random graphs [4], random Boolean formulae [5], and the partitioning of random sets of numbers [ibid.]. Consider n urns U [0] , U [1] , ..., U [n− 1] , into which we throw H = H (n) balls with equal probability to land in each urn; and associated binary r.v.’s, also denoted by U [i]. An empty urn is indicated