On Infinite-Difference Sets

Let A. be an infinite, strictly increasing sequence of non-negative 1 integers with d(A.) > 0 for i = I, ... ,h. Let the infinite difference set 1 D. of A. be the set of non-negative integers which occur infinitely often 1 1 as the difference of two terms of A .. This paper gives several results on 1 infinite difference sets, thereby answering some questions posed by Erdos. It follows from Theorems I and 2 that D1n ... nDh has positive lower density and does not contain gaps of arbitrary length. There exists even a sequence A with ~(A) > 0 whose infinite difference set equals D1n ... nDh. Theorem 4 says that the collection of infinite difference sets associated with sequences of positive upper density is a filter on the set of all subsets of the non-negative integers. It follows from Theorem 6 that an infinite difference set need not contain an infinite arithmetical progression. Theorems 7 and 8 are related to a problem of Motzkin. He asked how dense a sequence A can be if its difference set does not contain any elements from a given set K. It is a consequence of Theorem 8 that if k 1,k2 , •.• is a sequence of positive integers such that l· k./k. h <~for some positive integer h, J J J+ then there exists a sequence A with d(A) > 0 such that k. is not contained J in the difference set of A for j = 1,2, .•.• All proofs in the paper are elementary and self-contained. Further most results are quantitative; for example, in the cases above where it is stated that E_(A) > 0 we in fact give explicit lower bounds for E_(A).