Vojtěch Jarník (Czech pronunciation: [ˈvojcɛx ˈjarɲiːk]; 22 December 1897 – 22 September 1970) was a Czech mathematician. He worked for many years as a professor and administrator at Charles University, and helped found the Czechoslovak Academy of Sciences. He is the namesake of Jarník's algorithm for minimum spanning trees. Jarník worked in number theory, mathematical analysis, and graph algorithms. He has been called "probably the first Czechoslovak mathematician whose scientific works received wide and lasting international response". As well as developing Jarník's algorithm, he found tight bounds on the number of integer lattice points on convex curves, studied the relationship between the Hausdorff dimension of sets of real numbers and how well they can be approximated by rational numbers, and investigated the properties of nowhere-differentiable functions.
Education and career Jarník was born on 22 December 1897. He was the son of Jan Urban Jarník, a professor of Romance language philology at Charles University, and his older brother, Hertvík Jarník, also became a professor of linguistics. 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. 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. In 1921 he completed a doctoral degree (RNDr.) at Charles University with a dissertation on Bessel functions supervised by Petr, then returned to Charles University as Petr's assistant. 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. 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. Jarník supervised the dissertations of 16 doctoral students. Notable among these are Miroslav Katětov, a chess master who became rector of Charles University, Jaroslav Kurzweil, known for the Henstock–Kurzweil integral, Czech number theorist Bohuslav Diviš, and Slovak mathematician Tibor Šalát. He died on 22 September 1970, at the age of 72.
Contributions Although Jarník's 1921 dissertation, like some of his later publications, was in mathematical analysis, his main area of work was in number theory. He studied the Gauss circle problem and proved a number of results on Diophantine approximation, integer lattice point problems, and the geometry of numbers. He also made pioneering, but long-neglected, contributions to combinatorial optimization.
Number theory
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
3 2 π 3 L 2 / 3 + O ( L 1 / 3 ) {\displaystyle {\frac {3}{\sqrt[{3}]{2\pi }}}L^{2/3}+O(L^{1/3})}
points of the integer lattice. The O {\displaystyle O} 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. 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. 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
| x − p q | < 1 q k {\displaystyle \left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{k}}}}
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. Besicovitch used different methods than Jarník to prove it, and the result has come to be known as the Jarník–Besicovitch theorem.
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