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On Irregularities of Distribution and Approximate Evaluation of Certain Functions II

  • W. W. L. Chen
Part of the Progress in Mathematics book series (PM, volume 70)

Abstract

Let U =[0,1]. Suppose that g is a Lebesgue-integrable function, not necessarily bounded, in U 2, and that h is any function in U 2. Let P = P(N) be a distribution of N points in U 2 such that h(y) is finite for every y ∈ P. For x = (x1,x2) in U 2, let B(x) denote the rectangle consisting of all y = (y1,y2) in U 2 satisfying 0 < yl < x1 and 0 < y2 < x2, and write
$${\rm{Z}}\left[ {P;{\rm{h}}:B({\rm{x}})} \right] = \sum\limits_{y{\kern 1pt} \in {\kern 1pt} P{\kern 1pt} \cap B({\rm{x}})} {{\rm{h(y}}).}$$
(1)
Let μ denote the Lebesgue measure in U2, and write
$${\rm{D}}\left[ {P;{\rm{h;g}};B({\rm{x}})} \right] = {\rm{Z}}\left[ {P;{\rm{h}};B({\rm{x}})} \right] - {\rm{N}}\int\limits_{B({\rm{x}})} {{\rm{g}}({\rm{y}}){\rm{d}}\mu } .$$
(2)

Keywords

Number Theory Lebesgue Measure Auxiliary Function Analytic Number Point Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    [1].W. W. L. Chen, On irregularities of distribution and approximate evaluation of certain functions, to appear in Quarterly Journal of Mathematics (Oxford) 1985.Google Scholar
  2. [2]
    G. Halász, On Roth’s method in the theory of irregularities of point distributions, Recent progress in analytic number theory, vol. 2, pp. 79–94 (Academic Press, London, 1981).Google Scholar
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    K. F. Roth, On irregularities of distribution, Mathematika, 1 (1954), 73–79.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    W. M. Schmidt, Irregularities of distribution VII, Acta Arith., 21 (1972), 45–50.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • W. W. L. Chen
    • 1
  1. 1.Imperial CollegeLondonUK

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