On three problems for sequences

If ξ∈ (0,1) and A=a_n, n?? is a sequence of real numbers define S_n(ξ,A)∶=Σ{a_k∶:k=nξ]+1 to n}, n??, where x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence S_n (ξ,A), n??, for all ξ in an appropriate set K of real numbers, that the sequence a_n, n??, converges to zero. It was shown that such a conclusion is possible if K={ξ,1?ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:

1. does the convergence of S_n (ξ,A), n??, for a single irrational number ξ imply a_n→0.
2. does the convergence of S_n(ξ,A), n??, for finitely many rational numbers ξ∈ (0, 1) imply a_n→0.
3. does the convergence of S_n (ξ,A), n??, for all rational numbers ξ∈ (0,1) imply a_n→0?