Exact order of subsets of asymptotic bases in additive number theory

Abstract

Let A be an asymptotic basis of order h. Define I k(A)={F|F⫅A,|F|=k andA⧹F is a basis} where | F| indicates the number of elements in the finite set F. We denote by g( A) the exact order of A. Define ▪ An open problem in additive number theory is to calculate G k ( h). We prove in this paper that G k(h)⩾( 4 3 )( h k+1 k+1+O(h k) as h tends to infinity, which is an improvement of a result of Melvyn B. Nathanson (The exact order of subsets of additive bases, in “Proceedings, New York Number Theory Seminar, 1982,” Lecture Notes in Mathematics, Vol. 1052, pp. 273–277, Springer-Verlag, 1984) . On the other hand, we estimate G k ( h) as k tends to infinity for any fixed integer h. It is proved that G k ( h) has order of magnitude k h−1 as k tends to infinity for any fixed h ≥ 2.