Dirichlet's approximation theorem

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a> 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
, with 
 
 
 
 1
 ≤
 N
 
 
 {\displaystyle 1\leq N}
 
, there exist integers 
 
 
 
 p
 
 
 {\displaystyle p}
 
 and 
 
 
 
 q
 
 
 {\displaystyle q}
 
 such that 
 
 
 
 1
 ≤
 q
 ≤
 N
 
 
 {\displaystyle 1\leq q\leq N}
 
 and

|
 
 q
 α
 −
 p
 
 |
 
 ≤
 
 
 1
 
 ⌊
 N
 ⌋
 +
 1
 
 
 
 <
 
 
 1
 N
 
 
 .
 
 
 {\displaystyle \left|q\alpha -p\right|\leq {\frac {1}{\lfloor N\rfloor +1}}<{\frac {1}{N}}.}

Here 
 
 
 
 ⌊
 N
 ⌋
 
 
 {\displaystyle \lfloor N\rfloor }
 
 represents the <a href="/facts/Integer_part/ssehyMMf">integer part</a> of 
 
 
 
 N
 
 
 {\displaystyle N}
 
.
This is a fundamental result in <a href="/facts/Diophantine_approximation/WGVMuWF4">Diophantine approximation</a>, 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

|
 
 α
 −
 
 
 p
 q
 
 
 
 |
 
 <
 
 
 1
 
 q
 
 2
 
 
 
 
 
 
 {\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 <a href="/facts/Irrationality_measure/Ck8DKqpH">irrationality measure</a> at least 2.
The <a href="/facts/Thue%25E2%2580%2593Siegel%25E2%2580%2593Roth_theorem/ll7ubGmd">Thue–Siegel–Roth theorem</a> 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 <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a> 
 
 
 
 (
 1
 +
 
 
 5
 
 
 )
 
 /
 
 2
 
 
 {\displaystyle (1+{\sqrt {5}})/2}
 
 can be much more easily verified to be inapproximable beyond exponent 2.

Dirichlet's approximation theorem open-in-new

Dirichlet's approximation theorem