Many results are asymptotic in the sense that certain bounds are true for sufficiently large values of the parameters involved. Sometimes, one can determine the explicit bounds for how large we need the quantities to be by carefully working through the estimates and some other times, we just know that there is a finite number after which the results hold but the arguments are “ineffective” and it’s impossible to determine bounds. Ineffectivity arises from if the arguments are of the following form: A \implies B and not A also implies B.
Examples of Ineffective results:
- Siegel Zero problem, and Siegel–Walfisz theorem on primes in arithmetic progressions.
- Lower bounds of class numbers and
- Roth’s theorem on Diophantine approximation. (Finding the number of approximate solutions is effective, the sizes of the solutions is not effective)
- Siegel’s theorem on integral points, Falting’s theorem: Even here the number of solutions is effective, the sizes are not.
Although the bounds of class number is still ineffective, the class number one problem to determine imaginary quadratic fields is solved using the effective arguments involving Baker’s theorems in Diophantine approximation.
A Neighbourhood Of Infinity 