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Exponential Sums and Lattice Points
Author(s) -
Huxley M. N.
Publication year - 1990
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-60.3.471
Subject(s) - mathematics , rounding , exponential function , lattice (music) , exponent , square lattice , curvature , combinatorics , mathematical analysis , geometry , statistical physics , physics , linguistics , philosophy , computer science , acoustics , ising model , operating system
The area A inside a simple closed curve C can be estimated graphically by drawing a square lattice of sides 1/ M . The number of lattice points inside C is approximately AM 2 . If C has continuous non‐zero radius of curvature, the number of lattice points is accurate to order of magnitude at most M α for any α < ⅔. We show that if the radius of curvature of C is continuously differentiate, then the exponent ⅔ may be replaced by7 11, extending the result of Iwaniec and Mozzochi [ 4 ] in which C was a circle. On the way we obtain results on two‐dimensional exponential sums, the average rounding error of the values of a smooth function at equally spaced arguments, and the number of lattice points close to a smooth arc.