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Professor Jarńık ’ s contributions to the Geometry of Numbers

1. Among the numerous articles of Vojtech Jarńık on Analysis and Number Theory there is a small set of papers written in the decade from 1939 to 1949 belonging to Geometric Number Theory, consisting of the papers (65,72,74,76,77,78). In the following the main results from these papers will be described and discussed, but before doing so some personal remarks seem appropriate. One of my first papers deals with the product of non-homogeneous linear forms. Being a self-confident young man then, I submitted it to Acta Arithmetica. After some months I was shocked to get a big envelope from the editor and thought that the paper was refused. Hours later I dared to open it and was very pleased to see that it contained a friendly report of the referee along with a list of his suggestions for changes. They were readily inserted into the manuscript. Most probably the referee was Professor Jarńık. He pointed out many minor errors and masterly corrected my German. Several years later I wrote a paper on successive minima. Before submitting it I consulted very carefully the Zentralblatt für Mathematik and the Mathematical Reviews and found to my considerable disappointment that one main result had been published by Jarńık (76) some 20 years earlier and so it never was published. (Incidently, a more general version of my second result in this paper was published later on by Barnes.) Then in November 1967 I had the privilege to meet Professor Jarńık. In a lecture at the Institute of Mathematics of the University of Vienna, entitled “Über einige Ergebnisse der Gitterpunktlehre”, he gave an outline of results of Bretislav Novák on the number of integer lattice points in ellipsoids. I still remember his clear and concise way of delivering the lecture, his precise use of German with a slight Czech accent and his dry humor.