Diophantine approximation

Diophantine approximation studies how closely numbers can be approximated by rational or algebraic numbers subject to bounds on their arithmetic complexity.

Transcendence theory proves that particular numbers are not algebraic numbers, or that several numbers satisfy no nonzero polynomial relation with algebraic coefficients. A recurring method constructs an auxiliary expression which is simultaneously too small analytically and too large arithmetically if the proposed values were algebraic.

Let $f_1,\dots ,f_s$ be meromorphic functions with $f_j^{\prime }\in K[f_1,\dots ,f_s]$ for a fixed number field $K$. Suppose $f_1,f_2$ are algebraically independent over $\mathbb{C}$ and have finite orders ${\rho }_1,{\rho }_2$. The set $S$ of points where every $f_j$ is finite and belongs to $K$ satisfies the displayed bound. The orders are growth orders for meromorphic functions. For $f_1=z$, $f_2=e^z$, the orders are zero and one and the functions are algebraically independent; an algebraic value of $e^{\alpha }$ at nonzero algebraic $\alpha$ would produce infinitely many such points $n\alpha$.

Suppose nonzero algebraic numbers ${\beta }_j$ form a Galois-stable set, with Galois-invariant integer weights $b_j$, and $k\ne 0$ is an integer. Choose $P\in \mathbb{Z}[X]$ vanishing at all ${\beta }_j$ with $P(0)\ne 0$, and a positive integer $c$ making all $c{\beta }_j$ algebraic integers. For a large prime $p$, put $D=(1+\mathrm{deg}P)p-1$ and $f_p(X)=c^D{X}^{p-1}P(X{)}^p/(p-1)!$, and let $F_p$ be the sum of its derivatives. The weighted sum $k{F}_p(0)+\sum b_j{F}_p({\beta }_j)$ is an integer, nonzero modulo $p$ when $p\nmid kcP(0)$. If the displayed exponential expression vanished, integrating $(e^{-z}{F}_p(z){)}^{\prime }=-e^{-z}{f}_p(z)$ along the finitely many segments to ${\beta }_j$ would bound that nonzero integer by $C^p/(p-1)!\to 0$. This contradiction proves the obstruction and supplies direct proofs for $e$ and $\pi$.

For algebraic $a\ne 0,1$ and algebraic irrational $b$, every value $\exp (b\lambda )$ with $e^{\lambda }=a$ is transcendental. Consequently any quotient of chosen nonzero logarithms of algebraic numbers is either rational or transcendental.

Assume $a$ and $a^b$ algebraic with algebraic irrational $b$. Apply Siegel lemma to construct a small-coefficient polynomial $P$ vanishing at the algebraic pairs $(s+tb,a^s(a^b{)}^t)$ on a large square grid. The function $P(z,e^{\lambda z})$ is a nonzero exponential polynomial. Its many initial zeros make later grid values exponentially tiny, whereas an algebraic norm estimate forbids such tiny nonzero values. Iterating extends the zero grid indefinitely. Its quadratic zero count in disks contradicts the linear zero-count growth of an exponential polynomial.

Mahler's method uses functional equations relating a function at $z$ and at $z^l$ to transport an auxiliary polynomial to points rapidly approaching zero. High vanishing order gives an upper bound which conflicts with an algebraic norm or height lower bound under assumed algebraicity of the values. Functional transcendence ensures that the auxiliary expression is not identically zero.

This lacunary series is analytic for $|z|<1$ and satisfies $f_l(z^l)=f_l(z)-z$. At roots of unity of $l$-power order, its radial tail diverges, giving a dense set of singularities on the unit circle and excluding algebraicity as a function. The Mahler method proves transcendence of its values at every algebraic $z$ with $0<|z|<1$. At $z=1/2$, reduced truncations have denominators $2^{l^m}$ and error less than twice their $-l$th power; for $l\ge 3$, the Roth theorem gives a shorter proof.

For $0<p<q$, truncating the Fredholm series at $n$ gives a rational with denominator at most $Q_n=q^{l^n}$ and positive error at most $2Q_n^{-l(1-\log p/\log q)}$ for large $n$. The displayed strict inequality first rules out a rational value by the rational separation bound, and then rules out an algebraic irrational value by Roth theorem. Reduced denominators are at most $Q_n$ and tend to infinity because the truncations are distinct. In particular $l\ge 3$ and $p<q^{\delta }$, $\delta <1/3$, imply transcendence.

A nonzero integer polynomial of bidegree at most $N$ can cancel the first $(N+1{)}^2-1$ Taylor coefficients of the Fredholm series. Functional transcendence makes the resulting analytic function nonzero. If $\alpha$ and $f_l(\alpha )$ were both algebraic, the relation $f_l({\alpha }^{l^r})=f_l(\alpha )-{\sum }_{j<r}{\alpha }^{l^j}$ would keep the auxiliary values in one fixed number field. Their Absolute logarithmic Weil height is at most $N{l}^{r}h(\alpha )(1+1/(l-1))+O_N(r+1)$, whereas their chosen absolute value is $O_N(|\alpha {|}^{T_N{l}^r})$. The Liouville height inequality forces $T_N=O(N)$, contradicting its quadratic lower bound for sufficiently large $N$.

In the modern strict sense an E-function satisfies a nonzero linear differential equation over $\overline{\mathbb{Q}}(z)$, has algebraic coefficients $a_n$ with all conjugates bounded by $C^{n+1}$, and has common positive denominators $d_n\le C^{n+1}$ making $d_n{a}_0,\dots ,d_n{a}_n$ algebraic integers. Siegel's original arithmetic growth condition permits, for each $ϵ>0$, bounds of size $O_{ϵ}(n^{ϵn})$ instead. The strict and original growth formulations should be distinguished rather than asserted equivalent. Exponentials with algebraic exponent satisfy the stronger conditions and a first-order differential equation.

Work in the compositum of the two coefficient fields. Conjugate coefficient bounds add, while multiplying the common denominators clears both coefficient lists. For Siegel's original bounds $O_{ϵ}(n^{ϵn})$, use $ϵ/2$ in each denominator estimate. A nonzero linear differential equation for the sum follows because its derivatives lie in the sum of the two finite-dimensional derivative spaces over the rational function field. The stricter exponential arithmetic bounds are preserved as well.

Let a vector of E-functions satisfy $f^{\prime }=A(z)f$ with $A\in {\mathrm{Mat}}_s(\overline{\mathbb{Q}}(z))$. At a nonzero algebraic point $\xi$ which is not a pole of $A$, the transcendence degree of the values over $\overline{\mathbb{Q}}$ equals the transcendence degree of the functions over $\overline{\mathbb{Q}}(z)$. In particular algebraically independent functions give algebraically independent values. The exponential system $f_j=e^{{\gamma }_{j}z}$, for rationally independent algebraic ${\gamma }_j$, recovers the Lindemann–Weierstrass theorem by evaluating at one.

For distinct algebraic numbers ${\alpha }_1,\dots ,{\alpha }_s$, the numbers $e^{{\alpha }_1},\dots ,e^{{\alpha }_s}$ are linearly independent over $\overline{\mathbb{Q}}$. Equivalently, exponentials of numbers linearly independent over $\mathbb{Q}$ are algebraically independent over $\overline{\mathbb{Q}}$. The special case of a nonzero algebraic exponent gives a transcendental exponential. In particular $e^{i\pi }=-1$ forces $\pi$ to be transcendental.

Under an assumed algebraicity of $i\pi$, form the product ${\prod }_j(1+e^{{\alpha }_j})$ over its conjugates; its one known vanishing factor makes the whole product zero and regroup the subset sums as $k+\sum b_j{e}^{{\beta }_j}=0$, with nonzero exponents and Galois-invariant integral weights. After clearing their denominators, use $f_p(z)=(cz{)}^{p-1}{\prod }_j(cz-c{\beta }_j{)}^p/(p-1)!$. The sum of its derivatives at each ${\beta }_j$ is divisible by $p$ in the algebraic integers, whereas at zero its first nonzero derivative is a fixed nonzero residue for all but finitely many primes. Summing with the weights gives a nonzero rational integer. Integrating the derivative of $e^{-z}\sum f_p^{(j)}(z)$ expresses that integer as a quantity bounded by $C^p/(p-1)!$, which tends to zero. This is the contradiction.

An irrational real number is badly approximable when $\Vert q\alpha \Vert \ge c/q$ for some $c>0$ and every positive integer $q$, where the double bars mean distance to the nearest integer. Equivalently every rational approximation obeys $|\alpha -a/q|\ge c/q^2$. The Diophantine bound for the square root of two gives a concrete example and supplies separated rotations for bilinear cancellation for badly approximable phases.

For the nearest integer $a$ to $q\sqrt{2}$, the nonzero integer $2q^2-a^2$ has absolute value at least one. Dividing by $q\sqrt{2}+a\le 4q$ proves the bound. Thus the points $0,\sqrt{2},\dots ,W\sqrt{2}$ modulo one are separated by at least $1/(4W)$. Ordering their distances from zero gives ${\sum }_{h\le W}\Vert h\sqrt{2}{\Vert }^{-1}\ll W\log (2W)$.

A vector $\omega \in {\mathbb{R}}^n$ satisfying $|k\cdot \omega |\ge \gamma |k{|}_1^{-\tau }$ for all nonzero integer vectors $k$, with $\gamma >0$. This excludes resonances and controls the small divisors in analytic cohomological equations on a Diophantine torus. Different choices of norm change the constant but not the nature of the condition.

A denominator such as $k\cdot \omega$ which may become small for large integer vectors $k$. Inverting a directional derivative on a torus divides the nonzero Fourier coefficients by $ik\cdot \omega$. A Diophantine condition bounds this loss polynomially, so exponential Fourier decay still gives convergence on a smaller complex strip.

For a universal constant $C>0$ and $0<\epsilon \le 1$, every real $\alpha$ admits $1\le n\le {\epsilon }^{-C}$ with distance to the nearest integer of $\alpha n^2$ less than $\epsilon$. A Fejér kernel detects failure of recurrence as a large quadratic exponential sum. The Van der Corput inequality for finite scalar sequences then produces a short linear near-return, whose suitable multiple gives the quadratic return. To include $\epsilon >1$, use the bound $\max \{1,{\epsilon }^{-C}\}$.

For a real number $x$, define $\Vert x{\Vert }_{\mathbb{R}/\mathbb{Z}}={\min }_{k\in \mathbb{Z}}|x-k|$. It lies in $[0,1/2]$ and obeys $\Vert \ell x\Vert \le |\ell |\Vert x\Vert$ for an integer $\ell$. It is a distance on the circle, rather than a norm on the real vector space.

If finitely many points on $\mathbb{R}/\mathbb{Z}$ are pairwise separated by distance modulo one at least $\delta >0$, then the displayed bound holds for $Q\ge 2$. An arc of length $2r$ contains at most $1+2r/\delta$ points. First sum over $\Vert {\theta }_j\Vert \le Q^{-1}$, then over the dyadic annuli $2^j/Q<\Vert {\theta }_j\Vert \le 2^{j+1}/Q$. Their contributions are $O(Q+{\delta }^{-1})$ and $O(Q{2}^{-j}+{\delta }^{-1})$, respectively. Summing proves the estimate. A point at zero contributes $Q$, using the convention $\min (Q,\infty )=Q$.

For every irrational number $\alpha$ and positive integer $N$, there are integers $p,q$ with $1\le q\le N$ such thatThe proof applies the pigeonhole principle to the $N+1$ fractional parts of $0,\alpha ,\dots ,N\alpha$.

A reduced fraction $p/q$ is a best rational approximation of the first kind to a real number $\alpha$ when no rational number with denominator at most $q$ is closer to $\alpha$.

Let $L_1,\dots ,L_n$ be linearly independent linear forms in $n$ variables with algebraic coefficients. For every $\epsilon >0$, the nonzero $\mathbf{x}\in {\mathbb{Z}}^n$ satisfyinglie in finitely many proper linear subspaces of ${\mathbb{Q}}^n$. There are variants involving a finite set of places of a number field.

If $\alpha$ is a real algebraic irrational number and $\epsilon >0$, thenhas only finitely many reduced rational solutions $p/q$.

Let $\alpha$ be an algebraic irrational number, let $S$ be a finite set of primes, and let $\epsilon >0$. There are only finitely many reduced fractions $p/q$ whose denominator is S-smooth number and which satisfyThis is the denominator-restricted consequence of the finite-place Schmidt subspace theorem.

Baker's theorem gives transcendence and quantitative nonvanishing results for such logarithmic forms.

A homogeneous linear form in logarithms of nonzero algebraic numbers has the form

If chosen logarithms of algebraic numbers are linearly independent over $\mathbb{Q}$, then they, together with one, are linearly independent over $\overline{\mathbb{Q}}$. More generally a nonzero homogeneous linear combination with algebraic coefficients is transcendental. Quantitative lower bounds for integer coefficients are a separate, effective aspect of logarithmic-form theory.

Choose logarithms of nonzero algebraic numbers ${\alpha }_1,\dots ,{\alpha }_n$ and algebraic numbers ${\beta }_0,\dots ,{\beta }_n$, and putLet $A_j$ bound $10$, the naive polynomial height of the minimal polynomial of ${\alpha }_j$, and $\exp (|\log {\alpha }_j|)$. Let $B$ bound all the corresponding heights of the ${\beta }_j$ and all $\log A_j$. There is an effective constant $C$, depending only on $n$ and the degree of the number field generated by the data, such that

This quantitative refinement belongs to the theory of Baker's theorem; its dependence on the scaled coefficient bound is more specific than that broad theorem.

Choose logarithms of nonzero algebraic numbers ${\alpha }_1,\dots ,{\alpha }_n$, putand choose $A_i\ge 10$ large enough to bound the naive polynomial height of the minimal polynomial of ${\alpha }_i$ and $\exp (|\log {\alpha }_i|)$. After relabelling the terms if useful, setThere is an effectively computable constant $C$, depending only on $n$ and the degree of the number field generated by the ${\alpha }_i$, such that

The division by $\log A_n$ is the useful refinement over the general lower bound for a linear form in logarithms when one algebraic number has a large height correlated with a coefficient.

For fixed positive $a,b$ and fixed nonzero $c$, with $x\ne y$, put $Y=\max (x,y)$ and take $\Lambda =\log (a/b)+n\log (x/y)$, swapping the variables and coefficients if necessary. The equation gives $\log |\Lambda |\le -n\log Y+O(1)$ when the larger-variable power is sufficiently large. A two-logarithm lower bound gives $\log |\Lambda |\ge -C(a,b)\log Y\log (2n)$. Hence $n\le C\log (2n)+O(1)$, so $n$ is bounded. The cases $x=y$ and a bounded larger-variable power are elementary. For each remaining exponent, the Thue theorem for a binomial form supplies finitely many pairs. The constant-power exception $x=y=1$, $a-b=c$, must be removed.

Let $L$ be a fixed number field and let $V\subset L$ be a fixed rational linear subspace for which the associated norm form has no unit-family degeneracy. There are effective constants $A,C>0$ such that every $z\in V\cap {\mathcal{O}}_L$ with nonzero field norm $m$ satisfiesTo prove this, factor the principal ideal $(z)$, write its generators as a bounded factor times powers of fundamental units, use the linear relations defining $V$, and apply the Baker lower bound for a homogeneous linear form in logarithms to bound the unit exponents by $O(\log |m|)$. For $L=\mathbb{Q}(\alpha )K$ with $[\mathbb{Q}(\alpha ):\mathbb{Q}]=3$, $[K:\mathbb{Q}]=2$, and $V={\mathrm{span}}_{\mathbb{Q}}\{1,\alpha ,\omega \}$ for a basis $1,\omega$ of $K$, the coprime degrees rule out the degeneracy.

Let $K$ be a number field of degree $D$, and let $L_1,\dots ,L_M$ be linear forms in $N>DM$ variables with coefficients in $K$ and projective height at most $\mathcal{H}\ge 1$. There is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ satisfying every $L_j(\mathbf{x})=0$ andFor $K=\mathbb{Q}$, this follows by applying the pigeonhole principle to the images of the integer box $\{0,\dots ,Y{\}}^N$ under $(L_1,\dots ,L_M)$. Expanding coefficients in a rational basis of $K$ gives the factor $D$ in the general count.

For $M<N$ homogeneous equations with integer coefficients of absolute value at most $A\ge 1$, map the box $\{0,\dots ,Q{\}}^N$ to its $M$ equation values. There are $(Q+1{)}^N$ inputs and at most $(2NAQ+1{)}^M$ outputs. Taking $Q=\lfloor (2NA{)}^{M/(N-M)}\rfloor$ makes the first count larger, since $2NAQ+1\le 2NA(Q+1)$. The difference of two inputs with the same output is the required nonzero bounded integer solution. Clearing denominators also treats rational equations; expanding a number-field basis replaces each field equation by finitely many rational equations.

The $j$th normalized derivative isIf $P(X)={\sum }_r{a}_r{X}^r$ has integer coefficients, then $D_{j}P(X)={\sum }_{r\ge j}\left(\genfrac{}{}{0ex}{}{r}{j}\right)a_r{X}^{r-j}$ also has integer coefficients. For $\mathrm{deg}P\le n$, its naive polynomial height is at most $2^{n}H(P)$.

For an algebraic number $\alpha$ of degree $d$ and a small $\kappa >0$, Siegel lemma constructs integer polynomials $P,Q$ of degree at most $n$ and exponentially bounded coefficients for which $P(X)+\alpha Q(X)$ has a zero at $X=\alpha$ of order close to $2n/d$. A Wronskian then prevents every specialization $P(X)+yQ(X)$ from having a high-order zero at a rational point. This is the auxiliary-polynomial mechanism in the Thue-Siegel method.

Let $F(X,Y)=P(X)+YQ(X)$ with $P,Q\in \mathbb{Z}[X]$ linearly independent, ${\mathrm{deg}}_{X}F\le n$, and $H(F)\le C^n$. For every $\delta >0$, if a reduced rational $p/q$ has sufficiently large denominator in terms of $C$ and $\delta$, then $F(X,y)$ has multiplicity at most $\delta n+1$ at $p/q$, for every fixed real $y$.

Indeed, the nonzero Wronskianhas degree below $2n$ and height at most $C_0^n$. A zero of multiplicity $r$ of $F(X,y)$ forces a zero of multiplicity at least $r-1$ of $W$. By Gauss lemma for polynomials, $(qX-p{)}^{r-1}$ then divides $W$ in $\mathbb{Z}[X]$, so $q^{r-1}$ divides the leading coefficient of $W$ and is at most $C_0^n$.

If a real algebraic number $\alpha$ has degree $d\ge 3$, then for every $\epsilon >0$ there are only finitely many reduced fractions $p/q$ such thatThe Thue-Siegel auxiliary polynomial proof chooses two such approximations, constructs an auxiliary polynomial nonzero at their pair, and compares its denominator lower bound with the upper bound supplied by its high-order zero at $(\alpha ,\alpha )$.

For a nonsquare positive integer $D$, the Pell equation isPeriodic continued fractions of $\sqrt{D}$ produce infinitely many integer solutions.

In a real quadratic integer ring, multiplying by a positive norm-one fundamental unit preserves a prescribed norm. Choose an exponent so that a positive representative lies in a half-open interval $[A,A\epsilon )$. Its conjugate is the prescribed norm divided by the representative, so both integer coefficients are bounded on that interval. Enumerating these reduced representatives gives a complete finite list of unit orbits of solutions.

A generalized Pell equation is $x^2-d{y}^2=N$ for a fixed nonsquare positive integer $d$ and nonzero integer $N$. Multiplication by units of a real quadratic number field preserves the norm up to sign, reducing solutions to finitely many unit orbits when an appropriate bounded reduction interval is used.

A simple continued fraction recursively expands a real number by taking integer parts and reciprocals. Its convergents are exceptionally good rational approximations.

For a quadratic irrational $\sqrt{D}$, write its complete quotient as $(\sqrt{D}+m_j)/d_j$. The recurrence $m_{j+1}=d_j{a}_j-m_j$, $d_{j+1}=(D-m_{j+1}^2)/d_j$, $a_{j+1}=\lfloor (\sqrt{D}+m_{j+1})/d_{j+1}\rfloor$ computes its continued fraction. For $D=19$, the successive pairs after the initial quotient are $(4,3),(2,5),(3,2),(3,5),(2,3),(4,1)$, and then the cycle repeats. They give the displayed period of length six.

The finite negative continued fraction isFor $a_i\ge 2$, it describes a negative-framed chain of unknots: the diagonal framings are $-a_i$ and adjacent linking numbers are one. Repeated slam-dunk moves give rational surgery coefficient $-[a_1,\dots ,a_r{]}^{-}$. The corresponding chain matrix is negative definite. To check this, the successive determinants of its negative have recurrence $d_j=a_j{d}_{j-1}-d_{j-2}$, starting with $d_0=1$, $d_1=a_1$; they are positive and increasing, so Sylvester's criterion applies.

A continued fraction $[a_0;a_1,a_2,\dots ]$ with integer $a_0$ and positive integer $a_j$ for $j\ge 1$. Its partial numerators are all one. The continued fraction convergents have the standard two-term numerator and denominator recurrences.

The map $T(x)=1/x-\lfloor 1/x\rfloor$ on $(0,1]$, with $T(0)=0$. Its digit $N(x)=\lfloor 1/x\rfloor$ gives the successive partial quotients $N(T^{j}x)$ of a continued fraction. This is distinct from the geometric Gauss map assigning surface normals.

On each branch with digit $a=\lfloor 1/x\rfloor$, the displayed extension of the Gauss continued-fraction map is invertible almost everywhere: $x=(a+x^{\prime }{)}^{-1}$ and $y=(y^{\prime }{)}^{-1}-a$, with $a=\lfloor 1/y^{\prime }\rfloor$. Its invariant joint probability density function is $[(\log 2)(1+xy{)}^2{]}^{-1}$. The branch Jacobian determinant is $[x^2(a+y{)}^2{]}^{-1}$, which cancels the transformation of $1+xy$. Integrating out $y$ gives the Gauss measure density.

The inverse branch of the $n$th iterate of the Gauss continued-fraction map on a fixed digit cylinder has the displayed form, with continued fraction convergents $p_j/q_j$. Its derivative has absolute value $(q_n+q_{n-1}y{)}^{-2}$. Since $q_{n-1}\le q_n$, the maximum-to-minimum derivative ratio is at most four. The density of the Gauss measure changes this ratio by at most a further factor two. The cylinders shrink to irrational points, so this uniform distortion bound and the Lebesgue density theorem prove ergodicity of the map.

The invariant measure for the Gauss continued-fraction map with density $1/[(\log 2)(1+x)]$ on $[0,1]$. Inverse branches $x=1/(n+y)$ give transformed density $(\log 2{)}^{-1}{\sum }_{n\ge 1}[(n+y{)}^{-1}-(n+y+1{)}^{-1}]$, which telescopes to the same density.

For a simple continued fraction, the complete quotients satisfy ${\theta }_0=x$, $a_n=\lfloor {\theta }_n\rfloor$ and ${\theta }_{n+1}=1/({\theta }_n-a_n)$ whenever the denominator is nonzero. Thus ${\theta }_n=[a_n;a_{n+1},\dots ]$.

Starting from $x_0=x>0$, definewhenever $x_n$ is not an integer. This produces $x=[a_0;a_1,a_2,\dots ]$.

A simple continued fraction terminates exactly when its value is rational. For a reduced rational $p/q$, each reciprocal step is one step of the Euclidean algorithm and replaces the denominator by a strictly smaller nonnegative remainder.

Truncating a simple continued fraction gives a convergent; sufficiently accurate rational approximations with bounded denominator occur among these convergents.

For nonsquare $N>1$, write the complete quotients after the initial term as $(\sqrt{N}+m)/d$. They exceed one, while their algebraic conjugates lie between minus one and zero. Thus $d<\sqrt{N}+m<2\sqrt{N}$. Comparing coefficients in $\sqrt{N}=(p_k{\alpha }_{k+1}+p_{k-1})/(q_k{\alpha }_{k+1}+q_{k-1})$ gives $p_k^2-N{q}_k^2=d_{k+1}(p_k{q}_{k-1}-q_k{p}_{k-1})$, and the last determinant has absolute value one. These small residues are useful candidates for smooth numbers in congruence-of-squares factorization.

A rational $p/q$ is a best approximation of the second kind to $\alpha$ when $|q\alpha -p|$ is no larger than $|b\alpha -a|$ for every competing rational $a/b$ with $0<b\le q$. Every simple continued-fraction convergent has this property.

With $M(a)=\left(\begin{array}{cc}a& 1\\ 1& 0\end{array}\right)$,

If $p_n/q_n=[a_0,\dots ,a_n]$, then appending a positive tail $x$ gives

Because each matrix $M(a)=\left(\begin{array}{cc}a& 1\\ 1& 0\end{array}\right)$ is symmetric, reversing the partial quotients transposes their product. If the convergents of $[a_0,\dots ,a_n]$ are $p_j/q_j$, then the last two convergents of $[a_n,\dots ,a_0]$ are

For positive $a_j,b_j$, define $p_0=a_0$, $p_{-1}=1$, $q_0=1$, $q_{-1}=0$ andThenand $p_n/q_n$ is the corresponding finite generalized continued fraction.

When $b_j=1$ and $a_j$ are positive integers, adjacent convergents satisfyEven and odd convergents are monotone from opposite sides, their separation is $1/(q_n{q}_{n+1})$, and $q_n\to \infty$. They therefore have a common limit $x$, with

If every $a_j=b_j=1$, then $p_n=F_{n+2}$ and $q_n=F_{n+1}$. Binet's formulashows that the convergents tend to $\varphi$ and gives sharp scaled errors through $F_{n+1}-\varphi F_n={\psi }^n$.

The Lagrange theorem says that a real number has an eventually periodic simple continued fraction exactly when it is a quadratic irrational.

The simple continued fraction of a quadratic irrational is eventually periodic. For a nonsquare positive integer $D$, the continued fraction of $\sqrt{D}$ is periodic after its integer part.

A continued fraction is purely periodic when its repeating block begins at its first partial quotient:

Let $\theta =[\overline{a_0,\dots ,a_n}]$, and let ${\theta }^{\prime }$ be its algebraic conjugate, the other fixed point of the Möbius transformation represented by $M(a_0)\cdots M(a_n)$. ThenThe identity follows by transposing the continued-fraction matrix and substituting $X=-1/Y$ in its fixed-point equation.

Successive complete quotients repeat after two steps:

Rationalizing successive complete quotients gives

If $p_n/q_n$ are the convergents of $\sqrt{11}$, then

For the even convergents of $\sqrt{11}$,Since the second factor has norm one, every such convergent satisfies $p_{2m}^2-11q_{2m}^2=-2$.

If an irrational algebraic number has degree $d$, then $|\alpha -p/q|>C(\alpha )q^{-d}$. Approximations of arbitrarily larger polynomial order therefore prove that a Liouville number is transcendental.

For an irrational $x$ with continued fraction convergents $p_n/q_n$, one has $|x-p_n/q_n|\le 1/(q_n{q}_{n+1})\le 1/(a_{n+1}{q}_n^2)$. The Liouville approximation theorem for a quadratic irrational gives a positive lower bound $C/q_n^2$ instead. Thus unbounded partial quotients preclude quadratic irrationality, and in particular imply $x^2$ is irrational.

For the degree-$d$ minimal polynomial $P\in \mathbb{Z}[X]$ of $\alpha$, the nonzero integer $q^{d}P(p/q)$ gives $|P(p/q)|\ge q^{-d}$. On a fixed neighbourhood of $\alpha$, the mean value theorem boundsOutside that neighbourhood the desired lower bound is immediate, giving $|\alpha -p/q|>c(\alpha )q^{-d}$ after decreasing the constant.

For distinct reduced fractions $p/q$ and $u/v$,If $q>v$, this is strictly greater than $q^{-2}$.

Every convergent series of positive rational terms has a subseries whose rational partial sums $p_j/q_j$ satisfyChoose each new term small enough to enforce all previous tail bounds. Separation of reduced fractions first makes $\alpha$ irrational, and the Liouville approximation theorem then rules out every finite algebraic degree.