Cassels Bases

This paper describes several classical constructions of thin bases of finite order in additive number theory, and, in particular, gives a complete presentation of a beautiful construction of J. W. S. Cassels of a class of polynomially asymptotic bases. Some open problems are also discussed. 1. Additive bases of finite order The fundamental object in additive number theory is the sumset. If h ≥ 2 and A1, . . . , Ah are sets of integers, then we define the sumset (1) A1 + · · ·+Ah = {a1 + · · ·+ ah : ai ∈ Ai for i = 1, . . . , h}. If A1 = A2 = · · · = Ah = A, then the sumset (2) hA = A+A+ · · ·+A } {{ } h summands is called the h-fold sumset of A. If 0 ∈ A, then A ⊆ 2A ⊆ · · · ⊆ hA ⊆ (h+ 1)A ⊆ · · · For example, {0, 1, 4, 5}+ {0, 2, 8, 10} = [0, 15] and {3, 5, 7, 11}+ {3, 5, 7, 11, 13, 17, 19}= {6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}. The set A is called a basis of order h for the set B if every element of B can be represented as the sum of exactly h not necessarily distinct elements of A, or, equivalently, if B ⊆ hA. The set A is an asymptotic basis of order h for B if the sumset hA contains all but finitely many elements of B, that is, if card(B\hA) < ∞. The set A is a basis (resp. asymptotic basis) of finite order for B if A is a basis (resp. asymptotic basis) of order h for B for some positive integer h. The set A of nonnegative integers is a basis of finite order for the nonnegative integers only if 0, 1 ∈ A. Many classical results and conjectures in additive number theory state that some “interesting” or “natural” set of nonnegative integers is a basis or asymptotic basis of finite order. For example, the Goldbach conjecture asserts that the set of odd prime numbers is a basis of order 2 for the even integers greater than 4. Lagrange’s theorem states the set of squares is a basis of order 4 for the nonnegative integers 2000 Mathematics Subject Classification. 11B13, 11B75, 11P70,11P99.