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The number of configurations in lattice point counting II
Author(s) -
Huxley M. N.,
Žunić Joviša
Publication year - 2013
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/pdt011
Subject(s) - mathematics , combinatorics , integer (computer science) , quadrangle , regular polygon , geometry , discrete mathematics , archaeology , computer science , history , programming language
A convex plane set S is discretized by first mapping the centre of S to a point ( u , v ), preserving orientation, enlarging by a factor t to obtain the image S ( t , u , v ) and then taking the discrete set J ( t , u , v ) of integer points in S ( t , u , v ). Let N ( t , u , v ) be the size of the ‘configuration’ J ( t , u , v ). Let L ( N ) be the number of different configurations (up to equivalence by translation) of size N ( t , u , v ) = N and let M ( N ) be the number of different configurations with 1 ⩽ N ( t , u , v ) ⩽ N . Then L ( N ) ⩽ 2 N −1, M ( N ) ⩽ N 2 , with equality if S satisfies the Quadrangle Condition, that no image S ( t , u , v ) has four or more integer points on the boundary. For the circle, which does not satisfy the Quadrangle Condition, we expect that L ( N ) should be asymptotic to 2 N , despite the numerical evidence.