Dirichlet's approximation theorem: Difference between revisions

Revision as of 21:33, 2 March 2024

Revision as of 18:42, 26 March 2024

Line 9:

:<math> 0<\left | \alpha -\frac{p}{q} \right | < \frac{1}{q^2} </math>

is satisfied by infinitely many integers ''p'' and ''q''. This shows that any irrational number has [[irrationality measure]] at least 2.

is satisfied by infinitely many integers ''p'' and ''q''. This shows that any irrational number has [[irrationality measure]] at least 2. This corollary also shows that the [[Thue–Siegel–Roth theorem]], a result in the other direction, provides essentially the tightest possible bound, in the sense that the bound on rational approximation of [[algebraic number]]s cannot be improved by increasing the exponent beyond 2. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the [[golden ratio]] <math>(1+\sqrt{5})/2</math> can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the [[irrationality measure]].

The [[Thue–Siegel–Roth theorem]] says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the [[golden ratio]] <math>(1+\sqrt{5})/2</math> can be much more easily verified to be inapproximable beyond exponent 2.

==Simultaneous version==