J van der Corput's plenary: ICM 1936
Johannes van der Corput was a plenary lecturer at the International Congress of Mathematicians in Oslo in July 1936 delivering the lecture Diophantische Approximationen on 17 July. We give an English translation of the Introduction to the lecture.
An excellent overview of the field of Diophantine approximations, from the oldest authors to the middle of last year, is given by Jurjen F Koksma's report, published six months ago, which contains a complete list of the literature. If he had completed the book now, and had not already done so in September of last year, he would have added all the chapters; because the past academic year was extremely fruitful and important for this theory. It is even more difficult for me to give you a complete overview of the discoveries of the past few months in the time allotted to me. After the excursion into the geometry of numbers, which you have just undertaken under the guidance of such an experienced guide [Louis J Mordell had given the lecture Minkowski's Theorems and Hypotheses on Linear Forms], I no longer need to speak about the Minkowski linear form theorem. I find it more difficult to ignore some other discoveries. I would like to tell you how this year, using the well-known Blichfeldt method, Minkowski's result has been tightened using the sums of powers of linear forms. If I had more time, I would tell you how a few months ago Kurt Mahler proved the strange Khinchin Transfer Theorem in a surprisingly simple way, and how Vojtech Jarnik came to the remarkable result that this Khinchin theorem cannot be tightened. The very latest studies on distribution functions, metric properties, approximations of irrational numbers by rational ones, I have to pass all of this tacitly. Of particular importance is a more recent result of Theodor Schneider that happily complements the well-known Thue-Siegel theorem.
Instead of going into this, as well as two analogous theorems by Kurt Mahler, I would like to draw your attention in this lecture to the investigation of functions modulo one. The most important methods here are the elementary Skolem method, the method of rhythmic functions and goniometric methods. Under very general conditions, Thoralf Skolem derived an upper bound for the number of grid points on and around given curves. I hope that a Groningen dissertation by Dirk Schepel will soon be published, in which various Skolem results will be generalized and tightened up. The goniometric methods are completely different. These lead to the investigation of functions modulo one by considering trigonometric sums ...
Last Updated April 2020