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We show that products of sufficiently thick Cantor sets generate trees in theplane with constant distance between adjacent vertices. Moreover, we prove thatthe set of choices for this distance has non-empty interior. We allow our treesto be countably infinite, which further distinguishes this work from previousresults on patterns in fractal sets. This builds on the authors' previous workon graphs and distance sets of products of Cantor sets of sufficient Newhousethickness.

Infinite constant gap length trees in products of thick Cantor sets