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Lattice Points on Similar Figures and Conics

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Abstract

Let us say that a plane figure F satisfies Steinhaus’ condition if for any positive integer n, there exists a figure F n similar to F which satisfies the condition \({|F_n\cap{\mathbb Z}^2|=n}\). For example, the circular disc satisfies Steinhaus’ condition. We prove that every compact convex region in the plane \({\mathbb R^2}\) satisfies Steinhaus’ condition. As for plane curves, it is known that the circle satisfies Steinhaus’ condition. We consider Steinhaus’ condition for other conics, and present several results.

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Correspondence to Takayasu Kuwata.

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Kuwata, T., Maehara, H. Lattice Points on Similar Figures and Conics. Graphs and Combinatorics 27, 441–450 (2011). https://doi.org/10.1007/s00373-011-1015-4

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