Diophantine Approximation - Measure of The Accuracy of Approximations

Measure of The Accuracy of Approximations

The obvious measure of the accuracy of a Diophantine approximation of a real number α by a rational number p/q is However, this quantity may always be made arbitrarily small by increasing the absolute values of p and q; thus the accuracy of the approximation is usually estimated by comparing this quantity to some function φ of the denominator q, typically a negative power of it.

For such a comparison, one may want upper bounds or lower bounds of the accuracy. A lower bound is typically described by a theorem like "for every element α of some subset of the real numbers and every rational number p/q, we have ". In some case, "every rational number" may be replaced by "all rational numbers except a finite number of them", which amounts to multiplying φ by some constant depending on α.

For upper bounds, one has to take into accounts that not all the "best" Diophantine approximations provided by the convergents may have the desired accuracy. Therefore the theorems take the form "for every element α of some subset of the real numbers, there are infinitely many rational numbers p/q such that ".