Exact order of subsets of asymptotic bases

Abstract Let A be an asymptotic basis of order h. Denote g(A) the smallest h such that A is an asymptotic basis of order h. Let Ik(A) = {F ∥ F ⊆ A, ∥F∥ = k, and AβF is an asymptotic basis}. Define G k (h)= max A g(A)⩽h max F∈I k (A) g(A⧹F) In this paper, we prove that G k (h) ≥ α k + 1 ( 2048 625 ) [ (k + 1) 4 ] ( h (k + 1) ) k + 1 + O(h k ) , where α k = 1 if k=0,1 ( mod 4) 4 3 if k=2( mod 4) 135 64 if k=3( mod 4) This improves a result of Xing-De Jia (1992, J. Number Theory41, 116–127).