Exponential Squared Integrability of the Discrepancy Function in Two Dimensions

Let AN be an N-point set in the unit square and consider the Discrepancy function DN(~x) ≔ ♯ ( AN ∩ [~0, ~x) ) −N|[~0, ~x)|, where ~x = (x1, x2) ∈ [0, 1], [0, ~x) = ∏2 t=1[0, xt), and |[~0, ~x)| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of (Schmidt, 1972) on the L∞ norm of DN. We show that necessarily ‖DN‖exp(Lα) & (logN)1−1/α , 2 ≤ α < ∞ . The case ofα = ∞ is the Theoremof Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we have ‖DN‖exp(Lα) . (logN)1−1/α , 2 ≤ α < ∞ , whenever N = 2 for some positive integer n. This estimate depends upon variants of the Chang-Wilson-Wolff inequality (Chang et al., 1985). We also provide similar estimates for the BMO norm of DN. 1. Main Theorems The common theme of the subject of irregularities of distribution is to show that, no matter how N points are selected, their distribution must be far from uniform. In the present article, we are primarily interested in the precise behavior of such estimates near the L∞ endpoint, phrased in terms of exponential Orlicz classes. We restrict our attention to the two-dimensional case. Let AN ⊂ [0, 1] be a set of N points in the unit square. For ~x = (x1, x2) ∈ [0, 1], we define the Discrepancy function associated toAN as follows: DN(~x) ≔ ♯ ( AN ∩ [0, ~x) ) −N|[0, ~x)| , where [0, ~x) is the axis-parallel rectangle in the unit square with one vertex at the origin and the other at ~x = (x1, x2), and |[0, ~x)| = x1 · x2 denotes the Lebesgue measure of the rectangle. This is the difference between the actual number of points in the rectangle [0, ~x) and the expected number of points in this rectangle. The relative size of this function, in All authors are grateful to the Fields Institute for hospitality and support, and to the National Science Foundation for support. 1 2 D. BILYK, M.T. LACEY, I. PARISSIS, AND A. VAGHARSHAKYAN various senses, must necessarily increase with N. The principal result in this direction is due to Roth (Roth, 1954): K. Roth’s Theorem. In all dimensions d ≥ 2, we have the following estimate (1.1) ‖DN‖2 & (logN)(d−1)/2 where the implied constant is only a function of dimension d. The same bound holds for the L norm, for 1 < p < ∞, (Schmidt, 1977b), and is known to be sharp as to the order of magnitude, see (Chen, 1980) and (Beck and Chen, 1987) for a history of this subject (for the case d = 2, see Corollary 1.5 below). The endpoint cases of p = 1 and p = ∞ are much harder. We concentrate on the case of p = ∞ in this note, just in dimension d = 2, and refer the reader to (Beck, 1989; Bilyk et al., 2008; Bilyk and Lacey, 2008; Halász, 1981) for more information about the case of d ≥ 3. For information about the case of p = 1, see (Halász, 1981; Lacey, 2006). As it has been shown in the fundamental theorem of W. Schmidt (Schmidt, 1972), in dimension d = 2, the lower bound on the L∞ norm of the Discrepancy function is substantially greater than the L estimate (1.1): W. Schmidt’s Theorem. For any setAN ⊂ [0, 1] we have (1.2) ‖DN‖∞ & logN . This theorem is also sharp: one particular example is the famous van der Corput set (van der Corput, 1935) – a detailed discussion is contained in §3. In this paper, we give an interpolant between the results of Roth and Schmidt, which is measured in the scale of exponential Orlicz classes. 1.3. Theorem. For any N-point setAN ⊂ [0, 1] we have ‖DN‖exp(Lα) & (logN)1−1/α , 2 ≤ α < ∞ . Of course the lower bound of (logN), the case of α = 2 above, is a consequence of Roth’s bound. The other estimates require proof, which is a variant of Halász’s argument (Halász, 1981). We give details below and also remark that this estimate in the context of the Small Ball Inequality (Talagrand, 1994; Temlyakov, 1995) is known (Dunker et al., 1998). In addition, we demonstrate that the previous theorem is sharp. 1.4. Theorem. For all N, there is a choice of AN, specifically the digit-scrambled van der Corput set (see Definition 3.5), for which we have ‖DN‖exp(Lα) . (logN)1−1/α , 2 ≤ α < ∞ . In view of Proposition 2.2, taking α = 2, the theorem above immediately yields the sharpness of the L lower bounds in d = 2 with explicit dependence of constants on p. 1.5. Corollary. For every 1 ≤ p < ∞, the setAN from Theorem 1.4 satisfies ‖DN‖p . p(logN), where the implied constant is independent of p. DISCREPANCY FUNCTION IN TWO DIMENSIONS 3 There is another variant of the Roth lower bound, which we state here. 1.6. Theorem. We have the estimate ‖DN‖BMO1,2 & (logN) , where the norm is the dyadic Chang-Fefferman product BMO norm (see Definition 2.11), introduced in (Chang and Fefferman, 1980). Indeed, this Theorem is just a corollary to a standard proof of Roth’s Theorem, and its main interest lies in the fact that the estimate above is sharp. It is useful to recall the simple observation that the BMO norm is insensitive to functions that are constant in either the vertical or horizontal direction. That is, we have ‖DN‖BMO1,2 = ‖D̃N‖BMO1,2 , where D̃N(x1, x2) = DN(x1, x2) − ∫ 1 0 DN(x1, x2) dx1 − ∫ 1 0 DN(x1, x2) dx2 + ∫ 1