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How to wedge very big balls into very small balls. (English) Zbl 0888.52014

Let \(B(m,r) \subset \mathbb{R}^n\) be an (open) ball with center \(m\in \mathbb{R}^n\) and of radius \(r>0\). If the coordinates of \(m\) and the radius \(r\) are integers (rational numbers), then \(B(m,r)\) is said to be integer (rational) ball. It is clear that for the family of rational balls the following holds: there is a finite number of such balls whose intersection contains \(m\) and is contained in \(B(m,r)\), if \(m,r\) are arbitrarily given.
The authors prove the analogous statement for integer balls, and they explore the minimum number of integer balls for which the respective theorem holds.

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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