On Cantor Sets Defined by Generalized Continued Fractions

We study a special class of generalized continuous fractions, both in real and complex settings, and show that in many cases, the set of numbers that can be represented by a continued fraction for that class form a Cantor set. Specifically, we study generalized continued fractions with a fixed absolute value and a variable coefficient sign. We ask the same question in the complex setting, allowing the coefficient's argument to be a multiple of \pi/2. The numerical experiments we conducted showed that in these settings the set of numbers formed by such continued fractions is a Cantor set for large values of the coefficient. Using an iterated function systems construction, we prove that this is true for both real and complex cases. We also observed that in some regimes (for absolute values of the coefficient smaller than two), the set forms a peculiar fractal, and we formulate some questions and conjectures on its properties. We expect that some restrictions on the coefficients of generalized continued fractions should lead to the appearance of Cantorvals (closed bounded sets that have dense interiors but contain no isolated points or intervals) or, in the complex case, two-dimensional analogs of Cantorvals. Our projects bring together topics from number theory, dynamical systems, fractal geometry, and complex analysis. We believe it can serve as a foundation for researchers to build upon in the future.