The area of empty axis-parallel boxes amidst 2-dimensional lattice points |
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| Institution: | 1. Universidad Nacional de Hurlingham, Instituto de Tecnología e Ingeniería, Av. Gdor. Vergara 2222 (B1688GEZ), Villa Tesei, Buenos Aires, Argentina;2. Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150 (B1613GSX), Los Polvorines, Buenos Aires, Argentina;3. National Council of Science and Technology (CONICET), Argentina;4. Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150 (B1613GSX), Los Polvorines, Buenos Aires, Argentina;1. Computer Science Department, Facultad de Ciencias Exactas y Naturales, University of Buenos Aires, Pabellón 0+Infinito, Ciudad Universitaria (C1428), Ciudad Autónoma de Buenos Aires, Argentina;2. Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain;3. Facultad de Ingeniería, University of Buenos Aires, Av. Paseo Colón 850 (C1063) Ciudad Autónoma de Buenos Aires, Argentina;4. Instituto de Ciencias, Universidad Nacional de General Sarmiento, J. M. Gutiérrez 1150 (B1613GSX) Los Polvorines, Provincia de Buenos Aires, Argentina |
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| Abstract: | The dispersion of a point set in the unit square is the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the lattice generators. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subrings of the ring of integers of quadratic fields. We have tight bounds for the dispersion of a lattice based on the largest continued fraction coefficient of the generators, accurate to within one half. We provide an equivalent formulation of Zaremba's conjecture. Using this framework we are able to give very short proofs of previous results. |
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| Keywords: | Dispersion Lattices Continued fractions |
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