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

3, \tfrac{5}{2}, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7}, \tfrac{30}{11}, \tfrac{49}{18}, \tfrac{68}{25}, \tfrac{87}{32}, \tfrac{106}{39}, \ldots\, .

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