Let C, D be integers with C^2 < D, then C/sqrt(D) is realized in five dimensions. Hint: let w = (1,0,0,0,0) and v have first component C. There may be a way to take this down to 4 dimensions, but I am tired. Anyway, a number-theoretic characterization should not be far away for lower dimensions.
UPDATE 2011.02.07 In d <= 4 dimensions, a similar construction works, for all positive integers C and k (given k is the sum of at most (d-1) squares) for C/sqrt(k + C^2), and in dimension d = 4 it is possible to cover many of the remaining cases with w = (1,1,0,0) or (1,1,1,0). Perhaps Will Jagy can tell us which quadratic irrationals stay out of A_4 ?
Gerhard "Ask Me About System Design" Paseman, 2011.01.25