Diophantine Approximation - Best Diophantine Approximations of A Real Number

Best Diophantine Approximations of A Real Number

Given a real number α, there are two ways to define a best Diophantine approximation of α. For the first definition, the rational number p/q is a best Diophantine approximation of α if

for every rational number p'/q' such that 0< q' ≤ q.

For the second definition, the above inequality is replaced by

A best approximation for the second definition is also a best approximation for the first one, but the converse is false.

The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction. For the first definition, one has to consider also the semiconvergents.

For example, the constant e = 2.718281828459045235... has the (regular) continued fraction representation

Its best approximations for the second definition are

while, for the first definition, they are