Diophantine approximation

The Diophantine approximation deals with the approximation of real numbers by rational numbers . The approximation is named after Diophantus of Alexandria .

The first task was the question of how well a real number can be approximated by rational numbers. For this task, a rational number a / b is a “good” approximation of a real number α if the absolute value of the difference a / b and α cannot be reduced if a / b is replaced with another rational fraction with a smaller denominator. The problem was solved in the 18th century by means of continuous fractions .

If the “best” approximations of a given number are known, the main task of the region is to search for the exact upper and lower bounds of the above-mentioned difference, expressed as a function of the denominator.

It seems that the boundaries depend on the nature of real numbers - the lower limit of the approximation of rational numbers by another rational number is greater than the lower limit of algebraic numbers , which is itself larger than the lower limit for real numbers. Thus, real numbers, which can be better approximated than the limit for algebraic numbers, are definitely transcendental numbers . This enabled Liouville in 1844 to receive the first explicitly given transcendental number. Later, using a similar method, it was proved that $\pi$ ${\ displaystyle \ pi}$ $\ pi$ and $e$ ${\ displaystyle e}$ $e$ are transcendental.

Thus, the Diophantine approximation and the theory of transcendental numbers are very close domains and have many common theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations .

Historical remarks

After Borel and Khinchin established that almost all numbers allow only “the worst approximation” by rational numbers, the direction of the metric theory of Diophantine approximations (the theory of approximations of independent quantities), which belongs to the classical branch of Diophantine approximations, was formed.

The new trend came from an unexpected side. By classifying transcendental numbers, Mahler formulated the main metric problem of the theory of transcendental numbers — the hypothesis of a “transcendence measure” of almost all numbers. When the hypothesis was proved, a deep connection between the classical theory of Diophantine approximations and the metric theory of transcendental numbers began to open. The result was the development of a new direction - the theory of approximations of dependent quantities.

In modern theory, there are three main approaches.

Global, studying the general laws of approximation. Examples of global statements are the Dirichlet and Kronecker theorems, the Minkowski conjecture on products of linear forms.

The individual approach concerns the properties of special numbers (algebraic numbers,

e,\pi ,\ln 2

{\ displaystyle e, \ pi, \ ln 2}

) or requires the construction of numbers with certain properties (Liouville numbers, Mahler’s T-numbers).

The metric approach is intermediate. The approach requires a description of the approximation properties of numbers based on the theory of measure ^[1] .

Top Diophantine approximations of real numbers

If a real number α is given , there are two ways to determine the best Diophantine approximation of α . In the first definition ^{[2], the} rational number p / q is the best Diophantine approximation of the number α , if

\left|\alpha -{\frac {p}{q}}\right|<\left|\alpha -{\frac {p'}{q'}}\right|,

{\ displaystyle \ left | \ alpha - {\ frac {p} {q}} \ right | <\ left | \ alpha - {\ frac {p '} {q'}} \ right |,}

for any rational number p ' / q' , other than p / q , such that 0 < q ′ ≤ q .

In the second definition ^[3] ^{[4], the} above inequality is replaced by

\left|q\alpha -p\right|<\left|q^{\prime }\alpha -p^{\prime }\right|.

{\ displaystyle \ left | q \ alpha -p \ right | <\ left | q ^ {\ prime} \ alpha -p ^ {\ prime} \ right |.}

The best approximation for the second definition is the best for the first definition, but the reverse is not true ^[5] .

The theory of continued fractions allows us to calculate the best approximation of a real number — for the second definition, fractions converge as ordinary continued fractions ^[4] ^[5] ^[6] . For the first definition, intermediate fractions should also be considered ^[2] .

  Note : We agree to denote by  ${\frac {p_{k}}{q_{k}}}$                 $p$         $k$             $q$         $k$                   ${\ displaystyle {\ frac {p_ {k}} {q_ {k}}}}$       suitable fractions of this continued fraction.  Fractional  ${\tfrac {p_{k-2}}{q_{k-2}}},{\tfrac {p_{k-2}+p_{k-1}}{q_{k-2}+q_{k-1}}},{\tfrac {p_{k-2}+2p_{k-1}}{q_{k-2}+2q_{k-1}}},...,{\tfrac {p_{k-2}+a_{k}p_{k-1}}{q_{k-2}+a_{k}q_{k-1}}}={\tfrac {p_{k}}{q_{k}}}$                   $p$         $k$       $-$       $2$             $q$         $k$       $-$       $2$                 $,$                 $p$         $k$       $-$       $2$           $+$         $p$         $k$       $-$       $one$                 $q$         $k$       $-$       $2$           $+$         $q$         $k$       $-$       $one$                   $,$                 $p$         $k$       $-$       $2$           $+$       $2$         $p$         $k$       $-$       $one$                 $q$         $k$       $-$       $2$           $+$       $2$         $q$         $k$       $-$       $one$                   $,$       $.$       $.$       $.$       $,$                 $p$         $k$       $-$       $2$           $+$         $a$         $k$             $p$         $k$       $-$       $one$                 $q$         $k$       $-$       $2$           $+$         $a$         $k$             $q$         $k$       $-$       $one$                   $=$               $p$         $k$             $q$         $k$                     ${\ displaystyle {\ tfrac {p_ {k-2}} {q_ {k-2}}}, {\ tfrac {p_ {k-2} + p_ {k-1}} {q_ {k-2} + q_ {k-1}}}, {\ tfrac {p_ {k-2} + 2p_ {k-1}} {q_ {k-2} + 2q_ {k-1}}, ..., {\ tfrac {p_ {k-2} + a_ {k} p_ {k-1}} {q_ {k-2} + a_ {k} q_ {k-1}}} = {\ tfrac {p_ {k}} {q_ {k}}}}$       for an even k , they form an increasing, and for an odd k - a decreasing sequence.  The extreme members of this sequence are suitable fractions of the same parity.  Intermediate terms between them are called intermediate fractions ^[7] .

For example, the constant e = 2,718281828459045235 ... has the representation in the form of a continuous fraction

[2;1,2,1,1,4,1,1,6,1,1,8,1,\ldots \;].

{\ displaystyle [2; 1,2,1,1,4,1,1,6,1,1,8,1, \ ldots \;].}

Her best performances by the second definition

3,{\tfrac {8}{3}},{\tfrac {11}{4}},{\tfrac {19}{7}},{\tfrac {87}{32}},\ldots \,,

{\ displaystyle 3, {\ tfrac {8} {3}}, {\ tfrac {11} {4}}, {\ tfrac {19} {7}}, {\ tfrac {87} {32}}, \ ldots \ ,,}

While by first definition the best views would be

3,{\tfrac {5}{2}},{\tfrac {8}{3}},{\tfrac {11}{4}},{\tfrac {19}{7}},{\tfrac {49}{18}},{\tfrac {68}{25}},{\tfrac {87}{32}},{\tfrac {106}{39}},\ldots \,.

{\ displaystyle 3, {\ tfrac {5} {2}}, {\ tfrac {8} {3}}, {\ tfrac {11} {4}}, {\ tfrac {19} {7}}, { \ tfrac {49} {18}}, {\ tfrac {68} {25}}, {\ tfrac {87} {32}, {\ tfrac {106} {39}}, \ ldots \ ,.}

A measure of the accuracy of approximations

An obvious measure of the accuracy of the Diophantine approximation of a real number α, a rational number p / q is $\left|\alpha -{\frac {p}{q}}\right|$ ${\ displaystyle \ left | \ alpha - {\ frac {p} {q}} \ right |}$ . However, this value can always be made arbitrarily small by increasing the absolute values of p and q . For this reason, the accuracy of the approximation is usually compared with some function φ of the denominator q , usually the negative degree of the denominator.

For such an assessment, you can use the upper limit of the lower bounds of accuracy. The lower bound is usually described by a theorem, like “For any element α of some subset of real numbers and any rational number p / q we have $\left|\alpha -{\frac {p}{q}}\right|>\phi (q)$ ${\ displaystyle \ left | \ alpha - {\ frac {p} {q}} \ right |> \ phi (q)}$ ". In some cases, “any rational number” can be replaced by “all rational numbers, with the exception of a finite number”, and this number is taken into account by multiplying φ by some constant depending on α .

For the upper bounds, one can take into account the fact that not all the “best” Diophantine approximations obtained when constructing a continued fraction can give the desired accuracy. Therefore, the theorems take the form “For any element α of a certain subset of real numbers, there are infinitely many rational numbers p / q such that $\left|\alpha -{\frac {p}{q}}\right|<\phi (q)$ ${\ displaystyle \ left | \ alpha - {\ frac {p} {q}} \ right | <\ phi (q)}$ ".

Badly approximating numbers

A poorly approximated number is the number x , for which there exists a positive constant c , such that for all rational p / q we have

\left|{x-{\frac {p}{q}}}\right|>{\frac {c}{q^{2}}}\ .

{\ displaystyle \ left | {x - {\ frac {p} {q}}} \ right |> {\ frac {c} {q ^ {2}} \.}

Poorly approximable numbers are exactly numbers with limited partial quotients ^[8] .

Lower bounds for Diophantine approximations

Approximation of rational numbers by other rational numbers

Rational number $\alpha ={\frac {a}{b}}$ ${\ displaystyle \ alpha = {\ frac {a} {b}}}$ can be obviously perfectly approximated by numbers ${\tfrac {p_{i}}{q_{i}}}={\tfrac {i\,a}{i\,b}}$ ${\ displaystyle {\ tfrac {p_ {i}} {q_ {i}}} = {\ tfrac {i \, a} {i \, b}}}$ for any positive whole i .

If a ${\tfrac {p}{q}}\not =\alpha ={\tfrac {a}{b}}\,,$ ${\ displaystyle {\ tfrac {p} {q}} \ not = \ alpha = {\ tfrac {a} {b}} \ ,,}$ we have

\left|{\frac {a}{b}}-{\frac {p}{q}}\right|=\left|{\frac {aq-bp}{bq}}\right|\geq {\frac {1}{bq}},

{\ displaystyle \ left | {\ frac {a} {b}} - {\ frac {p} {q}} \ right | = \ left | {\ frac {aq-bp} {bq}} \ right | \ geq {\ frac {1} {bq}},}

insofar as $|aq-bp|$ ${\ displaystyle | aq-bp |}$ is a positive integer and therefore not less than 1. This accuracy of approximation is bad relative to irrational numbers (see the next section).

It can be noted that the given proof uses a variant of the Dirichlet principle - a non-negative number not equal to 0, not less than 1. This obviously trivial remark is used in almost all proofs for the lower bounds of Diophantine approximations, even more complex ones.

Summing up, the rational number perfectly approaches itself, but it is badly approached by any other rational number.

Approximation of algebraic numbers, Liouville result

In the 1840s, Joseph Liouville obtained the first lower bound for the approximation of algebraic numbers — if x is an irrational algebraic number of degree n over rational numbers, then there is a constant c ( x )> 0 such that

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

{\ displaystyle \ left | x - {\ frac {p} {q}} \ right |> {\ frac {c (x)} {q ^ {n}}}}

for all integers p and q , where q > 0 .

This result allowed him to obtain the first proven example of a transcendental number, the Liouville constant

\sum _{j=1}^{\infty }10^{-j!}=0,110001000000000000000001000\ldots \,,

{\ displaystyle \ sum _ {j = 1} ^ {\ infty} 10 ^ {- j!} = 0.110001000000000000000001000 \ ldots \ ,,}

which does not satisfy the Liouville theorem, whichever degree n is chosen.

This relationship between Diophantine approximations and the theory of transcendental numbers has been observed to date. Many evidence techniques are common to these two areas.

Approximation of algebraic numbers, Thue-Siegel-Roth theorem

For more than a century, there have been many attempts to improve the Liouville theorem — any improvement in the boundary allows us to prove the transcendence of more numbers. The main improvements were made by Axel Thue ^[9] , K. L. Siegel ^[10] , Freeman Dyson ^[11] and K. F. Roth ^[12] , which eventually led to the Thue-Siegel-Roth theorem - If x is irrational algebraic number and ε , (small) positive real number, then there is a positive constant c ( x , ε ) , such that

\left|x-{\frac {p}{q}}\right|>{\frac {c(x,\varepsilon )}{q^{2+\varepsilon }}}

{\ displaystyle \ left | x - {\ frac {p} {q}} \ right |> {\ frac {c (x, \ varepsilon)} {q ^ {2+ \ varepsilon}}}}

for any integers p and q such that q > 0 .

In a sense, this result is optimal, since the assertion of the theorem is false for ε = 0. This is a direct consequence of the upper bounds described below.

Joint approximations of algebraic data

Subsequently, generalized this for the case of joint approximations, proving that if x ₁ , ..., x _n are algebraic numbers such that 1, x ₁ , ..., x _{n are} linearly independent rational numbers, and any positive real number ε is given , then there are only a finite number of rational n -tuples ( p ₁ / q , ..., p _n / q ) such that

|x_{i}-p_{i}/q|<q^{-(1+1/n+\varepsilon )},\quad i=1,\ldots ,n.

{\ displaystyle | x_ {i} -p_ {i} / q | <q ^ {- (1 + 1 / n + \ varepsilon)}, \ quad i = 1, \ ldots, n.}

Again, this result is optimal in the sense that it is impossible to remove ε from the exponent.

Effective Borders

All previous lower bounds are not , in the sense that the proof does not give a way to calculate the constant in the statement. This means that it is impossible to use the proof of the theorem to obtain the bounds of the solutions of the corresponding Diophantine equation. However, this technique can often be used to limit the number of solutions of such an equation.

Nevertheless, the improvement Feldman provides an effective boundary — if x is an algebraic number of degree n over rational numbers, then there are effectively computable constants c ( x )> 0 and 0 < d ( x ) < n such that

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

{\ displaystyle \ left | x - {\ frac {p} {q}} \ right |> {\ frac {c (x)} {| q | ^ {d (x)}}}}

performed for all rational numbers.

However, as with any effective version of the Baker theorem, the d and 1 / c constants are so large that this effective result cannot be applied in practice.

Upper bound for Diophantine approximations

Common upper bound

The first important result on upper bounds for Diophantine approximations is the Dirichlet approximation theorem , from which it follows that for any irrational number α there are infinitely many fractions ${\tfrac {p}{q}}\;$ ${\ displaystyle {\ tfrac {p} {q}} \;}$ such that

\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}\,.

{\ displaystyle \ left | \ alpha - {\ frac {p} {q}} \ right | <{\ frac {1} {q ^ {2}} \ ,.}

It immediately follows that it is impossible to get rid of ε in the statement of the Thue-Siegel-Roth theorem.

A few years later this theorem was improved to the following theorem of Borel (1903) ^[13] . For any irrational number α, there are infinitely many fractions. ${\tfrac {p}{q}}\;$ ${\ displaystyle {\ tfrac {p} {q}} \;}$ such that

\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}\,.

{\ displaystyle \ left | \ alpha - {\ frac {p} {q}} \ right | <{\ frac {1} {{\ sqrt {5}} q ^ {2}} \ ,.}

therefore ${\frac {1}{{\sqrt {5}}\,q^{2}}}$ ${\ displaystyle {\ frac {1} {{\ sqrt {5}} \, q ^ {2}}}}$ is the upper boundary of the Diophantine approximation of any irrational number. The constant in this result cannot be improved without excluding some irrational numbers (see below).

Equivalent Real Numbers

Definition : Two real numbers $x, y$ ${\ displaystyle x, y}$ are called equivalent ^[14] ^[15] if there are integers $a,b,c,d\;$ ${\ displaystyle a, b, c, d \;}$ with $ad-bc=\pm 1\;$ ${\ displaystyle ad-bc = \ pm 1 \;}$ such that:

y={\frac {ax+b}{cx+d}}\,.

{\ displaystyle y = {\ frac {ax + b} {cx + d}} \ ,.}

Equivalence is determined by an integral Möbius transformation over real numbers or a member of a modular group ${\text{SL}}_{2}^{\pm }(\mathbb {Z} )$ ${\ displaystyle {\ text {SL}} _ {2} ^ {\ pm} (\ mathbb {Z})}$ , a set of reversible 2 × 2 matrices over integers. Each rational number is equivalent to 0. Thus, rational numbers are the equivalence class of this relation.

This equivalence may encompass ordinary continued fractions, as the following Serre theorem shows:

Theorem : Two irrational numbers x and y are equivalent if and only if there are two positive integers h and k , such that when representing numbers x and y as continuous fractions

x=[u_{0};u_{1},u_{2},\ldots ]\,,

{\ displaystyle x = [u_ {0}; u_ {1}, u_ {2}, \ ldots] \ ,,}

y=[v_{0};v_{1},v_{2},\ldots ]\,,

{\ displaystyle y = [v_ {0}; v_ {1}, v_ {2}, \ ldots] \ ,,}

performed

u_{h+i}=v_{k+i}

{\ displaystyle u_ {h + i} = v_ {k + i}}

for any non-negative integer i . ^[sixteen]

Lagrange Spectrum

As stated above, the constant in the Borel theorem cannot be improved, as Hurwitz showed in 1891 ^[17] . Let be $\phi ={\tfrac {1+{\sqrt {5}}}{2}}$ ${\ displaystyle \ phi = {\ tfrac {1 + {\ sqrt {5}}} {2}}}$ - the golden ratio . Then for any real constant $c>{\sqrt {5}}\;$ ${\ displaystyle c> {\ sqrt {5}} \;}$ there are only finitely many rational numbers p / q such that

\left|\phi -{\frac {p}{q}}\right|<{\frac {1}{c\,q^{2}}}.

{\ displaystyle \ left | \ phi - {\ frac {p} {q}} \ right | <{\ frac {1} {c \, q ^ {2}}}.}

Therefore, improvement can only be obtained by excluding numbers equivalent to $\phi$ ${\ displaystyle \ phi}$ . More precisely ^[18] ^[19] : For any rational number $\alpha$ ${\ displaystyle \ alpha}$ which is not equivalent $\phi$ ${\ displaystyle \ phi}$ , there are infinitely many fractions ${\tfrac {p}{q}}\;$ ${\ displaystyle {\ tfrac {p} {q}} \;}$ such that

\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {8}}q^{2}}}.

{\ displaystyle \ left | \ alpha - {\ frac {p} {q}} \ right | <{\ frac {1} {{\ sqrt {8}} q ^ {2}}}.}

By successive elimination of equivalence classes - each should exclude numbers equivalent to ${\sqrt {2}}$ ${\ displaystyle {\ sqrt {2}}}$ - you can raise the lower limit. The values that can be obtained from this process are Lagrange numbers that are part . They converge to number 3 and are associated with Markov numbers ^[20] ^[21] .

Khinchin's theorem and its extensions

Let be $\psi$ ${\ displaystyle \ psi}$ is a non-increasing function of positive numbers to positive real numbers. The real number x (not necessarily algebraic) is called $\psi$ ${\ displaystyle \ psi}$ - approximable if there are infinitely many rational numbers p / q , such that ^[22]

\left|x-{\frac {p}{q}}\right|<{\frac {\psi (q)}{|q|}}.

{\ displaystyle \ left | x - {\ frac {p} {q}} \ right | <{\ frac {\ psi (q)} {| q |}}.}

Khinchin in 1926 proved that if the sequence $\sum _{q}\psi (q)$ ${\ displaystyle \ sum _ {q} \ psi (q)}$ is divergent, then almost all real numbers (in the sense of Lebesgue measure ) are $\psi$ ${\ displaystyle \ psi}$ -approximate, and in the case of sequence convergence almost any real number $\psi$ ${\ displaystyle \ psi}$ -approximate is not.

Duffin and Schaffer ^[23] proved a more general theorem, from which Khinchin’s result follows and expressed a hypothesis, now known as the ^[24] . Beresnevich and Velani ^[25] proved that the analogue of the Duffin – Schaffer hypothesis on the Hausdorff measure is equivalent to the original Duffin – Schaffer hypothesis, which is a priori weaker.

Hausdorff dimension of exceptional sets

An important example of the function $\psi$ ${\ displaystyle \ psi}$ , to which the Khinchin theorem can be applied, is the function $\psi _{c}(q)=q^{-c}$ ${\ displaystyle \ psi _ {c} (q) = q ^ {- c}}$ , where c > 1. For this function, the corresponding series converge, so that, by the Khinchin theorem, the set $\psi _{c}$ ${\ displaystyle \ psi _ {c}}$ -approximate numbers have zero Lebesgue measure on the real axis. The Jarnik - Bezikovich theorem states that the Hausdorff dimension of this set is equal to $1/c$ ${\ displaystyle 1 / c}$ ^[26] . In particular, the set of numbers $\psi _{c}$ ${\ displaystyle \ psi _ {c}}$ -approximate for some $c>1$ ${\ displaystyle c> 1}$ (known as very well approximable numbers ), has the dimension of one, while the set of numbers, $\psi _{c}$ ${\ displaystyle \ psi _ {c}}$ -approximate for all $c>1$ ${\ displaystyle c> 1}$ (known as Liouville numbers ), has a Hausdorff dimension of zero.

Another important example is the function $\psi _{\epsilon }(q)=\epsilon q^{-1}$ ${\ displaystyle \ psi _ {\ epsilon} (q) = \ epsilon q ^ {- 1}}$ where $\epsilon >0$ ${\ displaystyle \ epsilon> 0}$ . For this function, the corresponding sequences diverge, and, by the Khinchin theorem, almost all numbers $\psi _{\epsilon }$ ${\ displaystyle \ psi _ {\ epsilon}}$ -approximate. In other words, these numbers are well approximable (that is, they are not badly approximable). Thus, the analogue of the Jarnik-Besikovich theorem should relate to the Hausdorff dimension of poorly approximable numbers. And Jarnik, in fact, proved the equality to the unit of the Hausdorff dimension of the set of such numbers. This result improved , who showed that the set of poorly approximable numbers is incompressible in the sense that if $f_{1},f_{2},\ldots$ ${\ displaystyle f_ {1}, f_ {2}, \ ldots}$ Is a sequence of bilipschitz mappings, then the Hausdorff dimension of the set of numbers x for which all $f_{1}(x),f_{2}(x),\ldots$ ${\ displaystyle f_ {1} (x), f_ {2} (x), \ ldots}$ poorly approximable, equal to one. Schmidt summarized Yarnik's theorem to higher dimensions, which is a significant achievement, because Yarnik’s arguments using the system of continued fractions essentially rely on the one-dimensionality of space.

Uniform distribution

Another section under study is the theory of an . Take the sequence a ₁ , a ₂ , ... real numbers and consider their fractional parts . That is, more formally, consider the sequence in R / Z , which is cyclic (can be considered as a circle). For any interval I on the circle, we consider the fraction of elements up to some integer N lying inside the interval, and compare this value with the fraction of the circle occupied by interval I. Homogeneous distribution means that in the limit, as N grows, the proportion of hits in the interval tends to the 'expected' value. Weil proved the basic result that this is equivalent to the boundedness of the Weyl sums formed from the sequence. This shows that the Diophantine approximation is closely related to the general problem of mutual cancellation in Weyl sums (estimates of the remainder term), which appear in the analytical theory of numbers .

A topic related to uniform distribution is the topic of uneven distribution , which has a combinatorial nature.

Unsolved Issues

There are still just formulated, but unsolved problems of Diophantine approximations, for example hypothesis and the lone runner hypothesis . It is also unknown whether there are algebraic numbers with unbounded coefficients in the decomposition into a continuous fraction.

Recent Studies

At the plenary meeting of the International Congress of Mathematicians in Kyoto (1990), Gregory A. Margulis outlined a broad program based on ergodic theory that allows one to prove theoretical and numerical results using the dynamic and ergodic properties of actions of subgroups of semi-simple Lie groups . Work D.Ya. Kleinbok and G.A. Margulis (with co-authors) demonstrates the power of this new approach to the classical problems of Diophantine approximations. Notable achievements include Margulis’s proof tens of years ago with further extensions (Dani and Margulis, Eskin – Margulis – Moses), and Kleinbock and Margulis’s proof of Baker and Sprinjuk’s hypotheses on diophantine approximations on manifolds. Various generalizations of Khinchin ’s aforementioned results on metric Diophantine approximations were obtained using this method.

Notes

↑ Springjuk, 1977 , p. 4-5 Preface.
↑ ¹ ² Khinchin, 1978 , p. 32.
↑ Cassels, 1961 , p. ten.
↑ ¹ ² Leng, 1970 , p. nineteen.
↑ ¹ ² Khinchin, 1978 , p. 35
↑ Cassels, 1961 , p. 10–17.
↑ Khinchin, 1978 , p. 21-22.
↑ Bugeaud, 2012 , p. 245.
↑ Thue, 1909 .
↑ Siegel, 1921 .
↑ Dyson, 1947 .
↑ Roth, 1955 .
↑ Perron, 1913 , p. Chapter 2, Theorem 15.
↑ Hurwitz, 1891 , p. 284.
↑ Hardy, Wright, 1979 , p. Chapter 10.11.
↑ See Perron’s article ( Perron 1929 , Chapter 2, Theorem 23, p. 63)
↑ Hardy, Wright, 1979 , p. 164.
↑ Cassels, 1961 , p. 21.
↑ Hurwitz, 1891 .
↑ Cassels, 1961 , p. 29.
↑ See Michel Waldschmidt: Introduction to Diophantine methods of irrationality and transcendence , pp 24–26.
↑ Springjuk, 1977 , p. 9 Chapter I.
↑ Duffin, Schaeffer, 1941 .
↑ Springjuk, 1977 , p. 23.
↑ Beresnevich, Velani, 2006 .
↑ Bernik, Beresnevich, Götze, Kukso, 2013 , p. 24

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Links

Diophantine Approximation: historical survey . From Michel Waldschmidt.
Hazewinkel, Michiel, ed. (2001), "p / d032600" , Encyclopedia of Mathematics , Springer , ISBN 978-1-55608-010-4

[_72380f81b32de20e-1] Springjuk, 1977 , p. 4-5 Preface.

[_72a7ec05ee6257f6-2] ¹ ² Khinchin, 1978 , p. 32.

[_045975f476bf6461-3] Cassels, 1961 , p. ten.

[_098d1b371b6d5b2c-4] ¹ ² Leng, 1970 , p. nineteen.

[_72a7ec05ee6257f1-5] ¹ ² Khinchin, 1978 , p. 35

[_8ecee46e69c6f24a-6] Cassels, 1961 , p. 10–17.

[_8b44db9f763821df-7] Khinchin, 1978 , p. 21-22.

[_0c86e2c5c3498b38-8] Bugeaud, 2012 , p. 245.

[_7f1bc5adac1df404-9] Thue, 1909 .

[_d41b3e00d7e8d249-10] Siegel, 1921 .

[_b16a060db62e7f57-11] Dyson, 1947 .

[_ae755347dcbc714a-12] Roth, 1955 .

[_718c17afa7ef154c-13] Perron, 1913 , p. Chapter 2, Theorem 15.

[_aa040882e1b1d0b7-14] Hurwitz, 1891 , p. 284.

[_4f12de1ec9989226-15] Hardy, Wright, 1979 , p. Chapter 10.11.

[16] See Perron’s article ( Perron 1929 , Chapter 2, Theorem 23, p. 63)

[_0af90ffdee1ad087-17] Hardy, Wright, 1979 , p. 164.

[_045972f476bf5f09-18] Cassels, 1961 , p. 21.

[_8604f6a7977f565b-19] Hurwitz, 1891 .

[_045972f476bf5f01-20] Cassels, 1961 , p. 29.

[21] See Michel Waldschmidt: Introduction to Diophantine methods of irrationality and transcendence , pp 24–26.

[_9f947fb61b2e3a05-22] Springjuk, 1977 , p. 9 Chapter I.

[_a9575d501c31c0eb-23] Duffin, Schaeffer, 1941 .

[_a7444d8b9f18d310-24] Springjuk, 1977 , p. 23.

[_95c0309f86e03d16-25] Beresnevich, Velani, 2006 .

[_72f01b62d68065f6-26] Bernik, Beresnevich, Götze, Kukso, 2013 , p. 24