Premium
Geometry of the gauss map and lattice points in convex domains
Author(s) -
Brandolini L.,
Colzani L.,
Iosevich A.,
Podkorytov A.,
Travaglini G.
Publication year - 2001
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300014376
Subject(s) - mathematics , regular polygon , geometry , boundary (topology) , gauss , combinatorics , rotation (mathematics) , lattice (music) , domain (mathematical analysis) , planar , convex curve , curvature , mathematical analysis , convex body , convex hull , physics , quantum mechanics , acoustics , computer graphics (images) , computer science
Let Ω be a convex planar domain, with no curvature or regularity assumption on the boundary. Let N θ ( R ) = card { R Ω θ ∩ℤ 2 }, where Ω θ denotes the rotation of Ω by θ. It is proved that, up to a small logarithmic transgression, N θ( R ) = |Ω| R 2 + O ( R 2/3 ), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of Ω under the Gauss map is also obtained.