Abstract :
Let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if all sufficiently large integers belong to P(S). A is called an asymptotic basis of order r if every large integer is a sum or at most r integers taken from A. Let Qα colon, equals ([αn2] : α > 0; n = 1, 2,...). We give a necessary and sufficient condition for Qα being an asymptotic basis. Let ordR(A) be the restricted order of A. We prove that ordR(Qa/b) ≥ (log a − 1)/log b, a/b > 0, (a, b) = 1; b > 1 and there are infinitely many a, b for which ordR(Qa/b) ≤ 3log2a + 10. Same related questions are also considered.
