Dirichlet's approximation theorem: Difference between revisions

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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

:<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#irrationality_exponent|irrationality exponent]] at least 2.

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.

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==Simultaneous version==

The simultaneous version of the Dirichlet's approximation theorem states that given real numbers <math>\alpha_1, \ldots, \alpha_d</math> and a natural number <math>N</math> then there are integers <math>p_1, \ldots, p_d, q\in\Z,1\le q\leq N^d</math> such that <math>\left|\alpha_i-\frac{p_i}q \right| \le \frac1{qN^{1/d}}.</math><ref>Schmidt, p. 27 Theorem 1A1B</ref>

==Method of proof==

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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.<ref>http://jeff560.tripod.com/p.html for a number of historical references.</ref> The method extends to simultaneous approximation.<ref>{{Springer|id=d/d032940|title=Dirichlet theorem}}</ref>

'''Proof outline''': Let <math>\alpha</math> be an irrational number and <math>nN</math> be an integer. For every <math>k=0, 1, ..., N</math> we can write <math>k\alpha=m_k + x_k</math> such that <math>m_k</math> is an integer and <math>0\le x_k <1</math>.

One can divide the interval <math>[0, 1)</math> into <math>N</math> smaller intervals of measure <math>\frac{1}{N}</math>. Now, we have <math>N+1</math> numbers <math>x_0,x_1,...,x_N</math> and <math>N</math> intervals. Therefore, by the pigeonhole principle, at least two of them are in the same interval. We can call those <math>x_i,x_j</math> such that <math>i < j</math>. Now:

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Suppose ''α'', ''p'', ''q'' are such that <math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{2q^2}</math>, and assume that ''α'' > ''p''/''q''. Then we may write <math>\alpha - \frac{p}{q} = \frac{\theta}{q^2}</math>, where 0 < ''θ'' < 1/2. We write ''p''/''q'' as a finite continued fraction [''a''₀; ''a''₁, ..., ''a_n''], where due to the fact that each rational number has two distinct representations as finite continued fractions differing in length by one (namely, one where ''a_n'' = 1 and one where ''a_n'' ≠ 1), we may choose ''n'' to be even. (In the case where ''α'' < ''p''/''q'', we would choose ''n'' to be odd.)

Let ''p''₀/''q''₀, ..., ''p_n''/''q_n'' = ''p''/''q'' be the convergents of this continued fraction expansion. Set <math>\omega := \frac{1}{\theta} - \frac{q_{n-1}}{q_n}</math>, so that <math>\theta = \frac{q_n}{q_{n-1} + \omega q_n}</math> and thus,<math display="block">\alpha = \frac{p}{q} + \frac{\theta}{q^2} = \frac{p_n}{q_n} + \frac{1}{q_n (q_{n-1} + \omega q_n)} = \frac{(p_n q_{n-1} + 1) + \omega p_n q_n}{q_n (q_{n-1} + \omega q_n)} = \frac{p_{n-1} q_n + \omega p_n q_n}{q_n (q_{n-1} + \omega q_n)} = \frac{p_{n-1} + \omega p_n}{q_{n-1} + \omega q_n}, </math> where we have used the fact that ''p_n''_-1−1 ''q_n'' - ''p_n'' ''q_n''_-1−1 = (-1)''ⁿ'' and that ''n'' is even.

Now, this equation implies that ''α'' = [''a''₀; ''a''₁, ..., ''a_n'', ''ω'']. Since the fact that 0 < ''θ'' < 1/2 implies that ''ω'' > 1, we conclude that the continued fraction expansion of ''α'' must be [''a''₀; ''a''₁, ..., ''a_n'', ''b''₀, ''b''₁, ...], where [''b''₀; ''b''₁, ...] is the continued fraction expansion of ''ω'', and therefore that ''p_n''/''q_n'' = ''p''/''q'' is a convergent of the continued fraction of ''α''.

{{collapse bottom}}

This theorem forms the basis for [[Wiener's attack]], a polynomial-time exploit of the [[RSA (cryptosystem)|RSA cryptographic protocol]] that can occur for an injudicious choice of public and private keys (specifically, this attack succeeds if the prime factors of the public key ''n'' = ''pq'' satisfy ''p'' < ''q'' < 2''p'' and the private key ''d'' is less than (1/3)''n''^1/4).<ref>{{cite journal|last=Wiener|first=Michael J.|date=1990|title=Cryptanalysis of short RSA secret exponents|url=https://ieeexplore.ieee.org/document/54902|journal=[[IEEE Transactions on Information Theory]]|volume=36|issue=3|pages=553–558|doi=10.1109/18.54902 |via=IEEE}}</ref>

==See also==