= Perron tree
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A Perron tree is a finite triangular splitting-and-sliding arrangement retaining a prescribed angular range while overlapping its pieces strongly. The compression parameter and the number of generations can make union area an arbitrarily small fraction of total piece area. Suitable longitudinal extensions separate the pieces into disjoint rectangles. This stronger finite geometry is used by the ball-multiplier counterexample; a compact measure-zero <Besicovitch set> alone does not assert that arbitrary segment translates are disjoint.
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