Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers formula_1 and formula_2, with formula_3, there exist integers formula_4 and formula_5 such that formula_6 and Here formula_8 represents the integer part of formula_2. This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality is satisfied by infinitely many integers "p" and "q". 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 numbers 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 formula_11 can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the irrationality measure. Simultaneous version. The simultaneous version of the Dirichlet's approximation theorem states that given real numbers formula_12 and a natural number formula_13 then  there are integers formula_14 such that formula_15 Method of proof. Proof By The Pigeonhole Principle. This theorem is a consequence of the pigeonhole principle. Peter Gustav Lejeune Dirichlet who proved the result used the same principle in other contexts (for example, the Pell equation) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later. The method extends to simultaneous approximation. Proof Outline: Let formula_16 be an irrational number and formula_17 be an integer. For every formula_18 we can write formula_19 such that formula_20 is an integer and formula_21. One can divide the  interval formula_22 into formula_17 smaller intervals of measure formula_24. Now, we have formula_25 numbers formula_26 and formula_17 intervals. Therefore, by the pigeonhole principle, at least two of them are in the same interval. We can call those formula_28 such that formula_29. Now: Dividing both sides by formula_31 will result in: And we proved the theorem. Proof By Minkowski's theorem. Another simple proof of the Dirichlet's approximation theorem is based on Minkowski's theorem applied to the set Since the volume of formula_34 is greater than formula_35, Minkowski's theorem establishes the existence of a non-trivial point with  integral coordinates. This proof extends naturally to simultaneous approximations by considering the set