Vojtěch Jarník

A convex curve through 13 integer lattice points

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

${\displaystyle {\frac {3}{\sqrt[{3}]{2\pi }}}L^{2/3}+O(L^{1/3})}$

points of the integer lattice. The ${\displaystyle O}$ 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.[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

${\displaystyle \left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{k}}}}$

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.[11] Besicovitch used different methods than Jarník to prove it, and the result has come to be known as the Jarník–Besicovitch theorem.[12]

Jarník's work in real analysis was sparked by finding, in the unpublished works of Bernard Bolzano, a definition of a continuous function that was nowhere differentiable.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]

Animation of Jarník's algorithm for minimum spanning trees

In computer science and combinatorial optimization, Jarník is known for an algorithm for constructing minimum spanning trees that he published in 1930, in response to the publication of Borůvka's algorithm by another Czech mathematician, Otakar Borůvka.[14] Jarník's algorithm builds a tree from a single starting vertex of a given weighted graph by repeatedly adding the cheapest connection to any other vertex, until all vertices have been connected. The same algorithm was later rediscovered in the late 1950s by Robert C. Prim and Edsger W. Dijkstra. It is also known as Prim's algorithm or the Prim–Dijkstra algorithm.[15]

He also published a second, related, paper with Miloš Kössler (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]