Homo (genus)

From Wikipedia, the free encyclopedia

Homo Temporal range: Piacenzian-Present, 2.1–0 Ma
Forensic reconstruction of an adult female Homo erectus
Scientific classification
Kingdom:	Animalia
Phylum:	Chordata
Class:	Mammalia
Order:	Primates
Suborder:	Haplorhini
Infraorder:	Simiiformes
Family:	Hominidae
Subfamily:	Homininae
Tribe:	Hominini
Genus:	Homo Linnaeus, 1758
Type species
Homo sapiens Linnaeus, 1758
Species
Homo sapiens †Homo erectus †Homo floresiensis †Homo habilis †Homo heidelbergensis †Homo naledi †Homo neanderthalensis

Homo (Latin homō "human being") is the genus that encompasses the extant species Homo sapiens (modern humans), plus several extinct species classified as either ancestral to or closely related to modern humans (depending on a species), most notably Homo erectus and Homo neanderthalensis.   The genus is taken to emerge with the appearance of Homo habilis, just more than two million years ago. Homo is derived from the genus Australopithecus, which itself had previously split from the lineage of Pan, the chimpanzees. Taxonomically, Homo is the only genus assigned to the subtribe Hominina which, with the subtribes Australopithecina and Panina, comprise the tribe Hominini.

Homo erectus appeared about two million years ago and, in several early migrations, it spread throughout Africa (where it is dubbed Homo ergaster) and Eurasia. It was likely the first human species to live in a hunter-gatherer society and to control fire. An adaptive and successful species, Homo erectus persisted for more than a million years, and gradually diverged into new species by around 500,000 years ago.

Homo sapiens (anatomically modern humans) emerges close to 300,000 to 200,000 years ago, most likely in Africa, and Homo neanderthalensis emerges at around the same time in Europe and Western Asia. H. sapiens dispersed from Africa in several waves, from possibly as early as 250,000 years ago, and certainly by 130,000 years ago, the so-called Southern Dispersal beginning about 70,000 years ago leading to the lasting colonisation of Eurasia and Oceania by 50,000 years ago. Both in Africa and Eurasia, H. sapiens met with and interbred with archaic humans. Separate archaic (non-sapiens) human species are thought to have survived until around 40,000 years ago (Neanderthal extinction), with possible late survival of hybrid species as late as 12,000 years ago (Red Deer Cave people).

Among extant populations of Homo sapiens, the deepest temporal division is found in the San people of Southern Africa, estimated at close to 130,000 years.

Names and taxonomy

Evolutionary tree chart emphasizing the subfamily Homininae and the tribe Hominini. After diverging from the line to Ponginae the early Homininae split into the tribes Hominini and Gorillini. The early Hominini split further, separating the line to Homo from the lineage of Pan. Currently, tribe Hominini designates the subtribes Hominina, containing genus Homo; Panina, genus Pan; and Australopithecina, with several extinct genera—the subtribes are not labelled on this chart.

A model of the evolution of the genus Homo over the last 2 million years (vertical axis). The rapid "Out of Africa" expansion of H. sapiens is indicated at the top of the diagram, with admixture indicated with Neanderthals, Denisovans, and unspecified archaic African hominins. Late survival of robust australopithecines (Paranthropus) alongside Homo until 1.2 Mya is indicated in purple.

The Latin noun homō (genitive hominis) means "human being" or "man" in the generic sense of "human being, mankind". The binomial name Homo sapiens was coined by Carl Linnaeus (1758). Names for other species of the genus were introduced beginning in the second half of the 19th century (H. neanderthalensis 1864, H. erectus 1892).

Even today, the genus Homo has not been properly defined. Since the early human fossil record began to slowly emerge from the earth, the boundaries and definitions of the genus Homo have been poorly defined and constantly in flux. Because there was no reason to think it would ever have any additional members, Carl Linnaeus did not even bother to define Homo when he first created it for humans in the 18th century. The discovery of Neanderthal brought the first addition.

The genus Homo was given its taxonomic name to suggest that its member species can be classified as human. And, over the decades of the 20th century, fossil finds of pre-human and early human species from late Miocene and early Pliocene times produced a rich mix for debating classifications. There is continuing debate on delineating Homo from Australopithecus—or, indeed, delineating Homo from Pan, as one body of scientists argue that the two species of chimpanzee should be classed with genus Homo rather than Pan. Even so, classifying the fossils of Homo coincides with evidences of: 1) competent human bipedalism in Homo habilis inherited from the earlier Australopithecus of more than four million years ago, (see Laetoli); and 2) human tool culture having begun by 2.5 million years ago.

From the late-19th to mid-20th centuries, a number of new taxonomic names including new generic names were proposed for early human fossils; most have since been merged with Homo in recognition that Homo erectus was a single and singular species with a large geographic spread of early migrations. Many such names are now dubbed as "synonyms" with Homo, including Pithecanthropus, Protanthropus, Sinanthropus, Cyphanthropus, Africanthropus, Telanthropus, Atlanthropus, and Tchadanthropus.

Classifying the genus Homo into species and subspecies is subject to incomplete information and remains poorly done. This has led to using common names ("Neanderthal" and "Denisovan") in even scientific papers to avoid trinomial names or the ambiguity of classifying groups as incertae sedis (uncertain placement)—for example, H. neanderthalensis vs. H. sapiens neanderthalensis, or H. georgicus vs. H. erectus georgicus. Some recently extinct species in the genus Homo are only recently discovered and do not as yet have consensus binomial names (see Denisova hominin and Red Deer Cave people). Since the beginning of the Holocene, it is likely that Homo sapiens (anatomically modern humans) have been the only extant species of Homo.

John Edward Gray (1825) was an early advocate of classifying taxa by designating tribes and families. Wood and Richmond (2000) proposed that Hominini ("hominins") be designated as a tribe that comprised all species of early humans and pre-humans ancestral to humans back to after the chimpanzee-human last common ancestor; and that Hominina be designated a subtribe of Hominini to include only the genus Homo—that is, not including the earlier upright walking hominins of the Pliocene such as Australopithecus, Orrorin tugenensis, Ardipithecus, or Sahelanthropus. Designations alternative to Hominina existed, or were offered: Australopithecinae (Gregory & Hellman 1939) and Preanthropinae (Cela-Conde & Altaba 2002); and later, Cela-Conde and Ayala (2003) proposed that the four genera Australopithecus, Ardipithecus, Praeanthropus, and Sahelanthropus be grouped with Homo within Hominina.

Evolution

Australopithecus

Forensic reconstruction of A. afarensis

Several species, including Australopithecus garhi, Australopithecus sediba, Australopithecus africanus, and Australopithecus afarensis, have been proposed as the direct ancestor of the Homo lineage. These species have morphological features that align them with Homo, but there is no consensus as to which gave rise to Homo.

Especially since the 2010s, the delineation of Homo from Australopithecus has become more contentious. Traditionally, the advent of Homo has been taken to coincide with the first use of stone tools (the Oldowan industry), and thus by definition with the beginning of the Lower Palaeolithic. But in 2010, evidence was presented that seems to attribute the use of stone tools to Australopithecus afarensis around 3.3 million years ago, close to a million years before the first appearance of Homo. LD 350-1, a fossil mandible fragment dated to 2.8 Mya, discovered in 2015 in Afar, Ethiopia, was described as combining "primitive traits seen in early Australopithecus with derived morphology observed in later Homo. Some authors would push the development of Homo close to or even past 3 Mya. Others have voiced doubt as to whether Homo habilis should be included in Homo, proposing an origin of Homo with Homo erectus at roughly 1.9 Mya instead.

The most salient physiological development between the earlier australopithecine species and Homo is the increase in endocranial volume (ECV), from about 460 cm³ (28 cu in) in A. garhi to 660 cm³ (40 cu in) in H. habilis and further to 760 cm³ (46 cu in) in H. erectus, 1,250 cm³ (76 cu in) in H. heidelbergensis and up to 1,760 cm³ (107 cu in) in H. neanderthalensis. However, a steady rise in cranial capacity is observed already in Autralopithecina and does not terminate after the emergence of Homo, so that it does not serve as an objective criterion to define the emergence of the genus.

Homo habilis

Forensic reconstruction of Homo habilis, exhibit in LWL-Museum für Archäologie, Herne, Germany (2007 photograph).

Homo habilis emerged about 2.1 Mya. Already before 2010, there were suggestions that H. habilis should not be placed in genus Homo but rather in Australopithecus. The main reason to include H. habilis in Homo, its undisputed tool use, has become obsolete with the discovery of Australopithecus tool use at least a million years before H. habilis. Furthermore, H. habilis was long thought to be the ancestor of the more gracile Homo ergaster (Homo erectus). In 2007, it was discovered that H. habilis and H. erectus coexisted for a considerable time, suggesting that H. erectus is not immediately derived from H. habilis but instead from a common ancestor. With the publication of Dmanisi skull 5 in 2013, it has become less certain that Asian H. erectus is a descendant of African H. ergaster which was in turn derived from H. habilis. Instead, H. ergaster and H. erectus appear to be variants of the same species, which may have originated in either Africa or Asia and widely dispersed throughout Eurasia (including Europe, Indonesia, China) by 0.5 Mya.

Homo erectus

Homo erectus has often been assumed to have developed anagenetically from Homo habilis from about 2 million years ago. This scenario was strengthened with the discovery of Homo erectus georgicus, early specimens of H. erectus found in the Caucasus, which seemed to exhibit transitional traits with H. habilis. As the earliest evidence for H. erectus was found outside of Africa, it was considered plausible that H. erectus developed in Eurasia and then migrated back to Africa. Based on fossils from the Koobi Fora Formation, east of Lake Turkana in Kenya, Spoor et al. (2007) argued that H. habilis may have survived beyond the emergence of H. erectus, so that the evolution of H. erectus would not have been anagenetically, and H. erectus would have existed alongside H. habilis for about half a million years (1.9 to 1.4 million years ago), during the early Calabrian.

A separate South African species Homo gautengensis has been postulated as contemporary with Homo erectus in 2010.

Dispersal

By about 1.8 million years ago, Homo erectus is present in both East Africa (Homo ergaster) and in Western Asia (Homo georgicus). The ancestors of Indonesian Homo floresiensis may have left Africa even earlier.
Homo erectus and related or derived archaic human species over the next 1.5 million years spread throughout Africa and Eurasia. Europe is reached by about 0.5 Mya by Homo heidelbergensis.
Homo neanderthalensis and Homo sapiens develop after about 300 kya. Homo naledi is present in Southern Africa by 300 kya.

H. sapiens soon after its first emergence spread throughout Africa, and to Western Asia in several waves, possibly as early as 250 kya, and certainly by 130 kya. Most notable is the Southern Dispersal of H. sapiens around 60 kya, which led to the lasting peopling of Oceania and Eurasia by anatomically modern humans. H. sapiens interbred with archaic humans both in Africa and in Eurasia, in Eurasia notably with Neanderthals and Denisovans. 

Among extant populations of Homo sapiens, the deepest temporal division is found in the San people of Southern Africa, estimated at close to 130,000 years, or possibly more than 300,000 years ago. Temporal division among non-Africans is of the order of 60,000 years in the case of Australo-Melanesians. Division of Europeans and East Asians is of the order of 50,000 years, with repeated and significant admixture events throughout Eurasia during the Holocene.

Archaic human species may have survived until the beginning of the Holocene (Red Deer Cave people), although they were mostly extinct or absorbed by the expanding H. sapiens populations by 40 kya.

List of species

The species status of H. rudolfensis, H. ergaster, H. georgicus, H. antecessor, H. cepranensis, H. rhodesiensis, H. neanderthalensis, Denisova hominin, Red Deer Cave people, and H. floresiensis remains under debate. H. heidelbergensis and H. neanderthalensis are closely related to each other and have been considered to be subspecies of H. sapiens.
There has historically been a trend to postulate "new human species" based on as little as an individual fossil. A "minimalist" approach to human taxonomy recognizes at most three species, Homo habilis (2.1–1.5 Mya, membership in Homo questionable), Homo erectus (1.8–0.1 Mya, including the majority of the age of the genus, and the majority of archaic varieties as subspecies, including H. heidelbergensis as a late or transitional variety) and Homo sapiens (300 kya to present, including H. neanderthalensis and other varieties as subspecies).

Diophantine approximation

From Wikipedia, the free encyclopedia

In number theory, the field of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. 

The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number a/b is a "good" approximation of a real number α if the absolute value of the difference between a/b and α may not decrease if a/b is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions.

Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator.

It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than the lower bound for algebraic numbers, which is itself larger than the lower bound for all real numbers. Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number. This allowed Liouville, in 1844, to produce the first explicit transcendental number. Later, the proofs that π and e are transcendental were obtained with a similar method.

Thus Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations.

Best Diophantine approximations of a real number

Given a real number α, there are two ways to define a best Diophantine approximation of α. For the first definition, the rational number p/q is a best Diophantine approximation of α if

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

for every rational number p'/q' different from p/q such that 0 < q′ ≤ q.

For the second definition, the above inequality is replaced by

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

A best approximation for the second definition is also a best approximation for the first one, but the converse is false.

The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction. For the first definition, one has to consider also the semiconvergents.

For example, the constant e = 2.718281828459045235... has the (regular) continued fraction representation

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

Its best approximations for the second definition are

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

while, for the first definition, they are

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

Measure of the accuracy of approximations

The obvious measure of the accuracy of a Diophantine approximation of a real number α by a rational number p/q is ${\displaystyle \left|\alpha -{\frac {p}{q}}\right|.}$ However, this quantity can always be made arbitrarily small by increasing the absolute values of p and q; thus the accuracy of the approximation is usually estimated by comparing this quantity to some function φ of the denominator q, typically a negative power of it.

For such a comparison, one may want upper bounds or lower bounds of the accuracy. A lower bound is typically described by a theorem like "for every element α of some subset of the real numbers and every rational number p/q, we have ${\displaystyle \left|\alpha -{\frac {p}{q}}\right|>\phi (q)}$ ". In some cases, "every rational number" may be replaced by "all rational numbers except a finite number of them", which amounts to multiplying φ by some constant depending on α.

For upper bounds, one has to take into account that not all the "best" Diophantine approximations provided by the convergents may have the desired accuracy. Therefore, the theorems take the form "for every element α of some subset of the real numbers, there are infinitely many rational numbers p/q such that ${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<\phi (q)}$ ".

Badly approximable numbers

A badly approximable number is an x for which there is a positive constant c such that for all rational p/q we have

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

The badly approximable numbers are precisely those with bounded partial quotients.

Lower bounds for Diophantine approximations

Approximation of a rational by other rationals

A rational number ${\displaystyle \alpha ={\frac {a}{b}}}$ may be obviously and perfectly approximated by ${\displaystyle {\tfrac {p_{i}}{q_{i}}}={\tfrac {i\,a}{i\,b}}}$ for every positive integer i.

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

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

because ${\displaystyle |aq-bp|}$ 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 pigeon hole 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

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 the rational numbers, then there exists a constant c(x) > 0 such that

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

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

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

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

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

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 Axel Thue (1909), Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955), leading finally to the Thue–Siegel–Roth theorem: If x is an irrational algebraic number and ε a (small) positive real number, then there exists a positive constant c(x, ε) such that

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

holds for every integer p and q such that q > 0.

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

Simultaneous approximations of algebraic numbers

Subsequently, Wolfgang M. Schmidt generalized this to the case of simultaneous approximations, proving that: If x₁, ..., x_n are algebraic numbers such that 1, x₁, ..., x_n are linearly independent over the rational numbers and ε is any given positive real number, then there are only finitely many rational n-tuples (p₁/q, ..., p_n/q) such that

${\displaystyle |x_{i}-p_{i}/q|$

Again, this result is optimal in the sense that one may not remove ε from the exponent.

Effective bounds

All preceding lower bounds are not 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) > 0 and 0 < d(x) < n such that

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

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.

Upper bounds for Diophantine approximations

General upper bound

The first important result about upper bounds for Diophantine approximations is Dirichlet's approximation theorem, which implies that, for every irrational number α, there are infinitely many fractions ${\displaystyle {\tfrac {p}{q}}\;}$ such that

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

This implies immediately that one cannot suppress the ε in the statement of Thue-Siegel-Roth theorem.

Over the years, this theorem has been improved until the following theorem of Émile Borel (1903). For every irrational number α, there are infinitely many fractions ${\displaystyle {\tfrac {p}{q}}\;}$ such that

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

Therefore, ${\displaystyle {\frac {1}{{\sqrt {5}}\,q^{2}}}}$ is an upper bound for the Diophantine approximations of any irrational number. The constant in this result may not be further improved without excluding some irrational numbers (see below).

Equivalent real numbers

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

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

So equivalence is defined by an integer Möbius transformation on the real numbers, or by a member of the Modular group ${\displaystyle {\text{SL}}_{2}^{\pm }(\mathbb {Z} )}$, the set of invertible 2 × 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an equivalence class for this relation.

The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of Serret:

Theorem: Two irrational numbers x and y are equivalent if and only there exist two positive integers h and k such that the regular continued fraction representations of x and y

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

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

verify

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

for every non negative integer i.

Thus, except for a finite initial sequence, equivalent numbers have the same continued fraction representation.

Lagrange spectrum

As said above, the constant in Borel's theorem may not improved, as shown by Adolf Hurwitz in 1891. Let ${\displaystyle \phi ={\tfrac {1+{\sqrt {5}}}{2}}}$ be the golden ratio. Then for any real constant c with ${\displaystyle c>{\sqrt {5}}\;}$ there are only a finite number of rational numbers p/q such that

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

Hence an improvement can only be achieved, if the numbers which are equivalent to ${\displaystyle \phi }$ are excluded. More precisely: For every irrational number ${\displaystyle \alpha }$, which is not equivalent to ${\displaystyle \phi }$, there are infinite many fractions ${\displaystyle {\tfrac {p}{q}}\;}$ such that

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

By successive exclusions — next one must exclude the numbers equivalent to ${\displaystyle {\sqrt {2}}}$ — of more and more classes of equivalence, the lower bound can be further enlarged. The values which may be generated in this way are Lagrange numbers, which are part of the Lagrange spectrum. They converge to the number 3 and are related to the Markov numbers.

Khinchin's theorem and extensions

Let ${\displaystyle \psi }$ be a non-increasing function from the positive integers to the positive real numbers. A real number x (not necessarily algebraic) is called ${\displaystyle \psi }$-approximable if there exist infinitely many rational numbers p/q such that

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

Aleksandr Khinchin proved in 1926 that if the series ${\displaystyle \sum _{q}\psi (q)}$ diverges, then almost every real number (in the sense of Lebesgue measure) is ${\displaystyle \psi }$-approximable, and if the series converges, then almost every real number is not ${\displaystyle \psi }$-approximable.

Duffin & Schaeffer (1941) proved a more general theorem that implies Khinchin's result, and made a conjecture now known by their name as the Duffin–Schaeffer conjecture. Beresnevich & Velani (2006) proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker.

Hausdorff dimension of exceptional sets

An important example of a function ${\displaystyle \psi }$ to which Khinchin's theorem can be applied is the function ${\displaystyle \psi _{c}(q)=q^{-c}}$, where c > 1 is a real number. For this function, the relevant series converges and so Khinchin's theorem tells us that almost every point is not ${\displaystyle \psi _{c}}$-approximable. Thus, the set of numbers which are ${\displaystyle \psi _{c}}$-approximable forms a subset of the real line of Lebesgue measure zero. The Jarník-Besicovitch theorem, due to V. Jarník and A. S. Besicovitch, states that the Hausdorff dimension of this set is equal to ${\displaystyle 1/c}$. In particular, the set of numbers which are ${\displaystyle \psi _{c}}$-approximable for some ${\displaystyle c>1}$ (known as the set of very well approximable numbers) has Hausdorff dimension one, while the set of numbers which are ${\displaystyle \psi _{c}}$-approximable for all ${\displaystyle c>1}$ (known as the set of Liouville numbers) has Hausdorff dimension zero.

Another important example is the function ${\displaystyle \psi _{\epsilon }(q)=\epsilon q^{-1}}$, where ${\displaystyle \epsilon >0}$ is a real number. For this function, the relevant series diverges and so Khinchin's theorem tells us that almost every number is ${\displaystyle \psi _{\epsilon }}$-approximable. This is the same as saying that every such number is well approximable, where a number is called well approximable if it is not badly approximable. So an appropriate analogue of the Jarník-Besicovitch theorem should concern the Hausdorff dimension of the set of badly approximable numbers. And indeed, V. Jarník proved that the Hausdorff dimension of this set is equal to one. This result was improved by W. M. Schmidt, who showed that the set of badly approximable numbers is incompressible, meaning that if ${\displaystyle f_{1},f_{2},\ldots }$ is a sequence of bi-Lipschitz maps, then the set of numbers x for which ${\displaystyle f_{1}(x),f_{2}(x),\ldots }$ are all badly approximable has Hausdorff dimension one. Schmidt also generalized Jarník's theorem to higher dimensions, a significant achievement because Jarník's argument is essentially one-dimensional, depending on the apparatus of continued fractions.

Uniform distribution

Another topic that has seen a thorough development is the theory of uniform distribution mod 1. Take a sequence a₁, a₂, ... of real numbers and consider their fractional parts. That is, more abstractly, look at the sequence in R/Z, which is a circle. For any interval I on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer N, and compare it to the proportion of the circumference occupied by I. Uniform distribution means that in the limit, as N grows, the proportion of hits on the interval tends to the 'expected' value. Hermann Weyl proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs throughout analytic number theory in the bounding of error terms.
Related to uniform distribution is the topic of irregularities of distribution, which is of a combinatorial nature.

Unsolved problems

There are still simply-stated unsolved problems remaining in Diophantine approximation, for example the Littlewood conjecture and the Lonely runner conjecture. It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.

Recent developments

In his plenary address at the International Mathematical Congress in Kyoto (1990), Grigory Margulis outlined a broad program rooted in ergodic theory that allows one to prove number-theoretic results using the dynamical and ergodic properties of actions of subgroups of semisimple Lie groups. The work of D.Kleinbock, G.Margulis, and their collaborators demonstrated the power of this novel approach to classical problems in Diophantine approximation. Among its notable successes are the proof of the decades-old Oppenheim conjecture by Margulis, with later extensions by Dani and Margulis and Eskin–Margulis–Mozes, and the proof of Baker and Sprindzhuk conjectures in the Diophantine approximations on manifolds by Kleinbock and Margulis. Various generalizations of the above results of Aleksandr Khinchin in metric Diophantine approximation have also been obtained within this framework.