An Algorithm for Packing Complements of F in Ite Sets of Integers

Let Ak = {0 = a1 < a2 < ... < ak} and B = {0 = b1 < b2 < ... < bn ...} be sets of k integers and infinitely many integers, respectively. Suppose B has asymptotic density x t d(B) x. If, for every integer n _> 0, there is at most one representation n a^ + bj , then we say that Ak has a packing complement of density j> x. Given Ak and x9 there is no known algorithm for determining whether or not B exists. We define "regular packing complement" and give an algorithm for determining if B exists when packing complement is replaced by regular packing complement. We exemplify with the case k = 5, i.e.s given A5 and x 1/10, we give an algorithm for determining if A5 has a regular packing complement B with density >_ 1/10. We relate this result to the Con/ecta/ie: Every A5 has a packing complement of density _> 1/10. Let Ak = {0 = a± < a2 < ... < ak} and B = {0 = b1 < b2 < ... < bn < ...} be sets of k integers and infinitely many integers, respectively. If, for every integer n >_ 0, n ai + bj has at most one solution, then we call B a packing complement, or p-complement, of Ak. Let B(n) denote the counting function of B and define d(B), the density of BJ as follows: d(B) = lim B(n)/n if this limit exists. From now on we consider only those sets B for which the density exists. For a given set Ak, we wish to find the p-complement B with maximum density. More precisely, we define p(Ak) s the packing codensity of Ak, as follows : p(Ak) = sup d(B) where B ranges over all p-complements of Ak. B Finally, we define p as the "smallest" p-codensity of any Ak, or, more precisely, pk =±nfp(Ak). We proved [1] that, for e > 0, G) l _ < p < 2 ^ 6 ^ + £ + i * y} if k is sufficiently large. The first four p are trivial, since we can find sets for which the lower bound is attained. Thus, A± = {0}, A2 = {0,1}, A3 = {0, 1, 3}, Ah = {0, 1, 4, 6}