Directional discrepancy in two dimensions
نویسندگان
چکیده
منابع مشابه
Directional Discrepancy in Two Dimensions
In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small ...
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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 < ∞ ...
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ژورنال
عنوان ژورنال: Bulletin of the London Mathematical Society
سال: 2011
ISSN: 0024-6093
DOI: 10.1112/blms/bdr050