Editing Diophantine approximation (section)

== Lower bounds for Diophantine approximations ==
{{unsourced section|date=May 2023}}
=== Approximation of a rational by other rationals ===
A rational number <math display="inline">\alpha =\frac{a}{b}</math> may be obviously and perfectly approximated by <math display="inline">\frac{p_i}{q_i} = \frac{i\,a}{i \,b}</math> for every positive integer ''i''.

If <math display="inline">\frac{p}{q} \not= \alpha = \frac{a}{b}\,,</math> we have
:<math>\left|\frac{a}{b} - \frac{p}{q}\right| = \left|\frac{aq - bp}{bq}\right| \ge \frac{1}{bq},</math>
because <math>|aq - bp|</math> is a positive integer and is thus not lower than 1. Thus the accuracy of the approximation is bad relative to irrational numbers (see next sections).

It may be remarked that the preceding proof uses a variant of the [[pigeonhole principle]]: a non-negative integer that is not 0 is not smaller than 1. This apparently trivial remark is used in almost every proof of lower bounds for Diophantine approximations, even the most sophisticated ones.

In summary, a rational number is perfectly approximated by itself, but is badly approximated by any other rational number.

=== Approximation of algebraic numbers, Liouville's result ===
{{main|Liouville number}}

In the 1840s, [[Joseph Liouville]] obtained the first lower bound for the approximation of [[algebraic number]]s: If ''x'' is an irrational algebraic number of degree ''n'' over the rational numbers, then there exists a constant {{nowrap|''c''(''x'') > 0}} such that

:<math> \left| x- \frac{p}{q} \right| > \frac{c(x)}{q^n}</math>

holds for all integers ''p'' and ''q'' where {{nowrap|''q'' > 0}}.

This result allowed him to produce the first proven example of a transcendental number, the [[Liouville constant]]
:<math>
\sum_{j=1}^\infty 10^{-j!} = 0.110001000000000000000001000\ldots\,,
</math>
which does not satisfy Liouville's theorem, whichever degree ''n'' is chosen.

This link between Diophantine approximations and transcendental number theory continues to the present day. Many of the proof techniques are shared between the two areas.

=== Approximation of algebraic numbers, Thue–Siegel–Roth theorem ===
{{main|Thue–Siegel–Roth theorem}}

Over more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound enables us to prove that more numbers are transcendental. The main improvements are due to {{harvs|first=Axel|last=Thue|authorlink=Axel Thue|year=1909|txt}}, {{harvs|frst=Carl Ludwig|last=Siegel|authorlink=Carl Ludwig Siegel|year=1921|txt}}, {{harvs|first=Freeman|last=Dyson|authorlink=Freeman Dyson|year=1947|txt}}, and {{harvs|first=Klaus|last=Roth|authorlink=Klaus Roth|year=1955|txt}}, leading finally to the Thue–Siegel–Roth theorem: If {{math|''x''}} is an irrational algebraic number and {{math|''ε > 0''}}, then there exists a positive real number {{math|''c''(''x'', ''ε'')}} such that
:<math>
    \left| x- \frac{p}{q} \right|>\frac{c(x, \varepsilon)}{q^{2+\varepsilon}}
</math>
holds for every integer {{math|''p''}} and {{math|''q''}} such that {{math|''q'' > 0}}.

In some sense, this result is optimal, as the theorem would be false with ''ε''&nbsp;=&nbsp;0. This is an immediate consequence of the upper bounds described below.

=== Simultaneous approximations of algebraic numbers ===
{{main|Subspace theorem}}
Subsequently, [[Wolfgang M. Schmidt]] generalized this to the case of simultaneous approximations, proving that: If {{math|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} are algebraic numbers such that {{math|1, ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} are [[linear independence|linearly independent]] over the rational numbers and {{math|''ε''}} is any given positive real number, then there are only finitely many rational {{math|''n''}}-tuples {{math|(''p''<sub>1</sub>/''q'', ..., ''p''<sub>''n''</sub>/''q'')}} such that
:<math>\left|x_i - \frac{p_i}{q}\right| < q^{-(1 + 1/n + \varepsilon)},\quad i = 1, \ldots, n.</math>

Again, this result is optimal in the sense that one may not remove {{math|''ε''}} from the exponent.

=== Effective bounds ===
All preceding lower bounds are not [[effective results in number theory|effective]], in the sense that the proofs do not provide any way to compute the constant implied in the statements. This means that one cannot use the results or their proofs to obtain bounds on the size of solutions of related Diophantine equations. However, these techniques and results can often be used to bound the number of solutions of such equations.

Nevertheless, a refinement of [[Baker's theorem]] by Feldman provides an effective bound: if ''x'' is an algebraic number of degree ''n'' over the rational numbers, then there exist effectively computable constants ''c''(''x'')&nbsp;>&nbsp;0 and 0&nbsp;<&nbsp;''d''(''x'')&nbsp;<&nbsp;''n'' such that

:<math>\left| x- \frac{p}{q} \right|>\frac{c(x)}{|q|^{d(x)}} </math>

holds for all rational integers.

However, as for every effective version of Baker's theorem, the constants ''d'' and 1/''c'' are so large that this effective result cannot be used in practice.