For any q? Yes. The hardest real number to approximate with rationals is (1+sqrt(5))/2. Its approximations that come closest are of the form fib(n+1)/fib(n)

For any q? Yes. The hardest real number to approximate with rationals is (1+sqrt(5))/2. Its approximations that come closest are of the form fib(n+1)/fib(n) where fib(n) is the n'th fibonacci number. The error converges to 1/(sqrt(5) fib(n)^2). None are within 1/(10 q^2). To learn more about this, read up on continued fractions.