For contraction ratios λ up to 2−√3 the circle meets Kλ×Kλ only in two corners; above it the intersection can be non-trivial or even a continuum, and Legendre symbols control intersections with 1/n².
Rational and non-trivial solutions to some algebraic equations in the Cantor sets
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On the intersection of Cantor set with the unit circle and some sequences
For contraction ratios λ up to 2−√3 the circle meets Kλ×Kλ only in two corners; above it the intersection can be non-trivial or even a continuum, and Legendre symbols control intersections with 1/n².