Finite and uniform stability of sphere coverings

By attaching cables to the centers of the balls and certain intersections of the boundaries of the balls of a ball covering ofE ^d with unit balls, we can associate to any ball covering a collection of cabled frameworks. It turns out that a finite subset of balls can be moved, maintaining the covering property, if and only if the corresponding finite subframework in one of the cabled frameworks is not rigid. As an application of this cabling technique we show that the thinnest cubic lattice sphere covering ofE ^d is not finitely stable. The first two authors were partially supported by the Hungarian National Science Foundation under Grant No. 326-0413.