Shatrovskĭi’s Construction of Thin Bases
نویسنده
چکیده
The set A of nonnegative integers is called a basis of order h if every nonnegative integer can be represented as the sum of exactly h not necessarily distinct elements of A. An additive basis A of order h is called thin if there exists c > 0 such that the number of elements of A not exceeding x is less than cx1/h for all x ≥ 1. This paper describes a construction of Shatrovskĭi of thin bases of order h. 1. Additive bases of finite order Let N0 denote the set of nonnegative integers, let h ≥ 2, and let (A1, . . . ,Ah) be an h-tuple of subsets of N0. We define the sumset A1 + · · ·+Ah = {a1 + · · ·+ ah : ai ∈ Ai for i = 1, . . . , h}. The h-fold sumset of a set A of nonnegative integers is hA = A+ · · ·+A } {{ } h terms . The h-tuple (A1, . . . ,Ah) is called an additive system of order h if A1 + · · ·+Ah = N0. The set A is an additive basis, or, simply, a basis, of order h if hA = N0. If (A1, . . . ,Ah) is an additive system of order h, then A = ⋃h i=1 Ai is a basis of order h. For example, let a1, . . . , a ′ h be relatively prime positive integers, and let Ai = a ′ i ∗N0 = {a ′ ivi : vi ∈ N0} for i = 1, . . . , h. There exists an integer C such that every integer n ≥ C can be represented in the form n = a1v1 + · · ·+ a ′ hvh with v1, . . . , vh ∈ N0 (Nathanson [8, Section 1.6], and so the set A = [0, C − 1] ∪ h ⋃
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