Editing Diophantine approximation (section)

=== Simultaneous approximations of algebraic numbers ===
{{main|Subspace theorem}}
Subsequently, [[Wolfgang M. Schmidt]] generalized this to the case of simultaneous approximations, proving that: If {{math|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} are algebraic numbers such that {{math|1, ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} are [[linear independence|linearly independent]] over the rational numbers and {{math|''ε''}} is any given positive real number, then there are only finitely many rational {{math|''n''}}-tuples {{math|(''p''<sub>1</sub>/''q'', ..., ''p''<sub>''n''</sub>/''q'')}} such that
:<math>\left|x_i - \frac{p_i}{q}\right| < q^{-(1 + 1/n + \varepsilon)},\quad i = 1, \ldots, n.</math>

Again, this result is optimal in the sense that one may not remove {{math|''ε''}} from the exponent.