Problems in Additive Number Theory, Iii: Thematic Seminars at the Centre De Recerca Matemàtica

This is a survey of open problems in different parts of combinatorial and additive number theory. 1. What sets are sumsets? Let N,N0, Z, and Z d denote, respectively, the sets of positive integers, nonnegative integers, integers, and d-dimensional integral lattice points. LetG denote an arbitrary abelian group and let X denote an arbitrary abelian semigroup, written additively. Let |S| denote the cardinality of the set S. For any sets A and B, we write A ∼ B if their symmetric difference is finite, that is, if |(A \B) ∪ (B \A)| < ∞. Definition 1. Let A and B be nonempty subsets of an additive abelian semigroup X. The most important definition in additive number theory is the sumset A + B of the sets A and B: A+B = {a+ b : a ∈ A, b ∈ B} . Let A + B = ∅ if A = ∅ or B = ∅. If h ≥ 3 and A1, . . . , Ah are subsets of X, then we construct the sumset A1 + · · ·+Ah inductively as follows: A1 + · · ·+Ah−1 +Ah = (A1 + · · ·+Ah−1) +Ah = {a1 + · · ·+ ah : ai ∈ Ai for all i = 1, . . . , h} . If A1 = A2 = · · · = Ah = A, then hA = A+ · · ·+A } {{ } h times is called the h-fold sumset of A. Definition 2. Let A be a nonempty subset of an additive abelian semigroup X. The set A is called a basis of order h for X if hA = X. The set A is called an asymptotic basis of order h for X if hA ∼ X. A basic problem is: What sets are sumsets? More precisely, Date: July 14, 2008. 2000 Mathematics Subject Classification. 11-02, 11B05, 11B13, 11B34, 11B75, 11A67, 11D04, 11D72, 11D85, 05B45, 05C38.