Some of Roth's ideas in Discrepancy Theory

We give a brief survey on some of the main ideas that Klaus Roth introduced into the study of irregularities of point distribution and, through a small selection of results, indicate how some of these ideas have been developed by him and others to obtain better understanding of this intriguing subject. 1. The Classical Problem Although parts of it border on harmonic analysis, combinatorics and probability theory, irregularities of point distribution began as a branch of the theory of uniform distribution, and may sometimes be described as a quantitative form of the theory. It originated from a conjecture of van der Corput [20, 21] in 1935 that expresses the fact that no infinite sequence in [0, 1] can, in a certain sense, be too evenly distributed; see [6, page 3] for a precise statement of the conjecture. This was confirmed in 1945 by van AardenneEhrenfest [1], who also gave a quantitative version [2] of this in 1949, with a relatively weak bound. In 1954, Roth [30] showed that van Aardenne-Ehrenfest’s quantitative version of the problem is equivalent to a geometric discrepancy problem concerning the distribution of a finite set of points in the unit square [0, 1]. We shall now describe the multidimensional version of this geometric discrepancy problem. In the sequel, the letter k will denote a positive integer greater than 1. Let P be a distribution of N points, not necessarily distinct, in the unit cube [0, 1]. For any point x = (x1, . . . , xk), let B(x) = [0, x1)× . . .× [0, xk) denote the rectangular box anchored at the origin and with opposite vertex x. Let Z[P;B(x)] denote the number of points of P that lie in B(x), and consider the discrepancy D[P;B(x)] = Z[P;B(x)]−Nx1 . . . xk. Roth showed that ∫ [0,1]k |D[P;B(x)]| dx k (logN)k−1, (1) 1The assumption that the boxes B(x) are half-open is introduced purely for convenience. 2Throughout, we adopt Vinogradov notation and . For any two functions f and g, we write f g to denote |f | 6 Cg for some positive absolute constant C. If f and g are non-negative, then we write f g to denote f > Cg for some positive absolute constant C. Furthermore, we use the notation and with subscripts; in this case, the implicit constant C may then depend on those subscripts. Any deviation from this convention will be indicated beforehand. ROTH’S IDEAS IN DISCREPANCY THEORY 151 from which it follows easily that sup x∈[0,1]k |D[P;B(x)]| k (logN)(k−1)/2. (2) Roth’s deduction of the inequality (1) contains two crucial ideas. For the benefit of the reader, we illustrate them in the special case k = 2. Trivial Discrepancy. Since the point set P is arbitrary, we have essentially no information about it, making it hard if not impossible to extract the discrepancy from those parts of the square [0, 1] that are near the points of P. On the other hand, those parts of the square that are short of points of P give rise to “trivial discrepancies” which we then exploit. One can make parts of the square deficient of points of P in a very simple way. If we partition the square into more than 2N subsets, then given that there are only N points, at least half of these subsets are devoid of points of P. More precisely, we can partition the square into similar rectangles of area 2−n, where the integer n is chosen to satisfy 2N 6 2 < 4N . Then there are 2 such rectangles, at least half of which contain no points of P. Roth then proceeded to extract the discrepancy from such “empty” rectangles. A typical rectangle of area 2−n under consideration is of the form B = [m121 , (m1 + 1)2−r1)× [m222 , (m2 + 1)2−r2) ⊂ [0, 1], (3) where m1,m2, r1, r2 are non-negative integers satisfying r1 + r2 = n. Consider the smaller rectangle of area 2−n−2 given by B′ = [m121 , (m1 + 12 )2 −r1)× [m222 , (m2 + 12 )2 −r2) ⊂ [0, 1], made up of the bottom left quarter of B. For any x = (x1, x2) ∈ B′, the rectangle B′(x) = [x1, x1 + 2−r1−1)× [x2, x2 + 2−r2−1) is similar to B′ and contained in B, and so does not contain any point of P if neither does B. In this case, the rectangle B′(x) has trivial discrepancy N2−n−2. A device to pick up this trivial discrepancy is given by the Rademacher function defined locally on such an empty rectangle B by Rr1,r2(x) =  +1 if x ∈ B′, −1 if x ∈ B′ + (2−r1−1, 0), −1 if x ∈ B′ + (0, 2−r2−1), +1 if x ∈ B′ + (2−r1−1, 2−r2−1), depending on which quadrant of B the point x falls into. Now ∫ B x1x2Rr1,r2(x) dx = ∫