Dirichlet's approximation theorem

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 {displaystyle alpha } and {displaystyle N} , with {displaystyle 1leq N} , there exist integers {displaystyle p} and {displaystyle q} such that {displaystyle 1leq qleq N} and {displaystyle left|qalpha -pright|leq {frac {1}{[N]+1}}<{frac {1}{N}}.} Here {displaystyle [N]} represents the integer part of {displaystyle N} . 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 {displaystyle left|alpha -{frac {p}{q}}right|<{frac {1}{q^{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 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 {displaystyle (1+{sqrt {5}})/2} can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the irrationality measure. Contents 1 Simultaneous version 2 Method of proof 2.1 Proof By The Pigeonhole Principle 2.2 Proof By Minkowski's theorem 3 See also 4 Notes 5 References 6 External links Simultaneous version The simultaneous version of the Dirichlet's approximation theorem states that given real numbers {displaystyle alpha _{1},ldots ,alpha _{d}} and a natural number {displaystyle N} then there are integers {displaystyle p_{1},ldots ,p_{d},qin mathbb {Z} ,1leq qleq N} such that {displaystyle left|alpha _{i}-{frac {p_{i}}{q}}right|leq {frac {1}{qN^{1/d}}}.} 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.[1] The method extends to simultaneous approximation.[2] Proof Outline: Let {displaystyle alpha } be an irrational number and {displaystyle n} be an integer. For every {displaystyle k=0,1,...,n} we can write {displaystyle kalpha =m_{k}+x_{k}} such that {displaystyle m_{k}} is an integer and {displaystyle 0leq x_{k}<1} . One can divide the interval {displaystyle [0,1)} into {displaystyle n} smaller intervals of measure {displaystyle {frac {1}{n}}} . Now, we have {displaystyle n+1} numbers {displaystyle x_{0},x_{1},...,x_{n}} and {displaystyle n} intervals. Therefore, by the pigeonhole principle, at least two of them are in the same interval. We can call those {displaystyle x_{i},x_{j}} such that {displaystyle i

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