On Roth’s Orthogonal Function Method in Discrepancy Theory

Motivated by the recent surge of activity on the subject, we present a brief non-technical survey of numerous classical and new results in discrepancy theory related to Roth’s orthogonal function method. Communicated by Yukio Ohkubo The goal of this paper is to survey the use of Roth’s orthogonal function method in the theory of irregularities of distribution, which, in a broad sense, means applications of orthogonal function decompositions to proving discrepancy estimates. The idea originated in a seminal paper of Klaus Roth [61, 1954] that, according to his own words “started a new theory” [26]. Since then and up to today this approach has been widely exploited to produce new important results in the the area. In the present expository article we trace the method from its origins to recent results (to which the author has made some contribution). It is our intention to keep the exposition concise, simple, and unobscured by the technicalities. We shall concentrate on the heuristics and intuition behind the underlying ideas, introduce classical and novel points of view on the method, as well as connections to other areas of mathematics. Before we proceed to the main part of the discussion, I would like to emphasize the tremendous influence that the aforementioned paper of Roth [61], entitled “On irregularities of distribution”, has had on the development of the field. Even the number of papers with identical or similar titles, that appeared in the subsequent years, attests to its importance: 4 papers by Roth himself (On irregularities of distribution. I–IV, [61, 62, 63, 64]), one by H. Davenport (Note on irregularities of distribution, [33]), 10 by W. M. Schmidt (Irregularities of distribution. I–IX, [65, 66, 67, 68, 69, 70, 71, 72, 73, 74]), 2 by J. Beck (Note on irregularities of distribution. I–II, [5, 6]), 4 by W. W. L. Chen 2010 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: 11K38, 11K06.