Geometrical sets with forbidden configurations

Given finite configurations, let us denote by the maximum density a set can have without containing congruent copies of any. We will initiate the study of this geometrical parameter, called the independence density of the considered configurations, and give several results we believe are interesting. For instance we show that, under suitable size and nondegeneracy conditions, progressively 'untangles' and tends to as the ratios between consecutive dilation parameters grow large; this shows an exponential decay on the density when forbidding multiple dilates of a given configuration, and gives a common generalization of theorems by Bourgain and by Bukh in geometric Ramsey theory. We also consider the analogous parameter in the more complicated framework of sets on the unit sphere, obtaining the corresponding results in this setting.

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