Jarník was born on 22 December 1897. He was the son of Jan Urban Jarník [cs], a professor of Romance languagephilology at Charles University,[2] and his older brother, Hertvík Jarník, also became a professor of linguistics.[3] Despite this background, Jarník learned no Latin at his gymnasium (the C.K. české vyšší reálné gymnasium, Ječná, Prague), so when he entered Charles University in 1915 he had to do so as an extraordinary student until he could pass a Latin examination three semesters later.[3]
He studied mathematics and physics at Charles University from 1915 to 1919, with Karel Petr as a mentor. After completing his studies, he became an assistant to Jan Vojtěch at the Brno University of Technology, where he also met Mathias Lerch.[3] In 1921 he completed a doctoral degree (RNDr.) at Charles University with a dissertation on Bessel functions supervised by Petr,[3] then returned to Charles University as Petr's assistant.[3][1][4]
While keeping his position at Charles University, he studied with Edmund Landau at the University of Göttingen from 1923 to 1925 and again from 1927 to 1929.[5] On his first return to Charles University he defended his habilitation, and on his return from the second visit, he was given a chair in mathematics as an extraordinary professor. He was promoted to full professor in 1935 and later served as Dean of Sciences (1947–1948) and Vice-Rector (1950–1953). He retired in 1968.[1][4]
The Gauss circle problem asks for the number of points of the integer lattice enclosed by a given circle.
One of Jarník's theorems (1926), related to this problem, is that any closed strictly convex curve with length L passes through
at most
points of the integer lattice. The in this formula is an instance of Big O notation. Neither the exponent of L nor the leading constant of this bound can be improved, as there exist convex curves with this many grid points.[8][9]
Another theorem of Jarník in this area shows that, for any closed convex curve in the plane with a well-defined length, the absolute difference between the area it encloses and the number of integer points it encloses is at most its length.[10]
Jarník also published several results in Diophantine approximation, the study of the approximation of real numbers by rational numbers.
He proved (1928–1929) that the badly approximable real numbers (the ones with bounded terms in their continued fractions) have Hausdorff dimension one. This is the same dimension as the set of all real numbers, intuitively suggesting that the set of badly approximable numbers is large. He also considered the numbers x
for which there exist infinitely many good rational approximations p/q, with
for a given exponent k > 2, and proved (1929) that these have the smaller Hausdorff dimension 2/k. The second of these results was later rediscovered by Besicovitch.[11] Besicovitch used different methods than Jarník to prove it, and the result has come to be known as the Jarník–Besicovitch theorem.[12]
Mathematical analysis
Jarník's work in real analysis was sparked by finding, in the unpublished works of Bernard Bolzano, a definition of a continuous function that was nowhere differentiable. Bolzano's 1830 discovery predated the 1872 publication of the Weierstrass function, previously considered to be the first example of such a function. Based on his study of Bolzano's function, Jarník was led to a more general theorem: If a real-valued function of a closed interval does not have bounded variation in any subinterval, then there is a dense subset of its domain on which at least one of its Dini derivatives is infinite. This applies in particular to the nowhere-differentiable functions, as they must have unbounded variation in all intervals. Later, after learning of a result by Stefan Banach and Stefan Mazurkiewicz that generic functions (that is, the members of a residual set of functions) are nowhere differentiable, Jarník proved that at almost all points, all four Dini derivatives of such a function are infinite. Much of his later work in this area concerned extensions of these results to approximate derivatives.[13]
He also published a second, related, paper with Miloš Kössler [cs] (1934) on the Euclidean Steiner tree problem. In this problem, one must again form a tree connecting a given set of points, with edge costs given by the Euclidean distance. However, additional points that are not part of the input may be added to make the overall tree shorter. This paper is the first serious treatment of the general Steiner tree problem (although it appears earlier in a letter by Gauss), and it already contains "virtually all general properties of Steiner trees" later attributed to other researchers.[7]
Recognition and legacy
Jarník was a member of the Czech Academy of Sciences and Arts, from 1934 as an extraordinary member and from 1946 as a regular member.[1] In 1952 he became one of the founding members of Czechoslovak Academy of Sciences.[1][4] He was also awarded the Czechoslovak State Prize in 1952.[1]
Jarníkova Street, the Jarníkova bus stop, and a commemorative sign honoring Jarník
^Beresnevich, Victor; Ramírez, Felipe; Velani, Sanju (2016), "Metric Diophantine approximation: Aspects of recent work", in Badziahin, Dzmitry; Gorodnik, Alexander; Peyerimhoff, Norbert (eds.), Dynamics and Analytic Number Theory: Proceedings of the Durham Easter School 2014, London Mathematical Society Lecture Note Series, vol. 437, Cambridge University Press, pp. 1–95, arXiv:1601.01948, doi:10.1017/9781316402696.002, ISBN978-1-316-40269-6, S2CID119304793. See Theorem 1.33 (the Jarník–Besicovitch theorem), p. 23, and the discussion following the theorem.
^Durnová, Helena (2004), "A history of discrete optimization", in Fuchs, Eduard (ed.), Mathematics Throughout the Ages, Vol. II, Prague: Výzkumné centrum pro dějiny vědy, pp. 51–184, ISBN9788072850464. See in particular page 127: "Soon after Borůvka's published his solution, another Czech mathematician, Vojtěch Jarník, reacted by publishing his own solution," and page 133: "Jarník’s article on this topic is an extract from a letter to O. Borůvka".
![{\displaystyle {\frac {3}{\sqrt[{3}]{2\pi }}}L^{2/3}+O(L^{1/3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0c68222b35e1df7945ea831dc936a38f4f70c1)
