Complex numbers with a prescribed order of approximation and Zaremba's conjecture

Abstract

Given $b=-A\pm i$ with A being a positive integer, we can represent any complex number as a power series in b with coefficients in $A=\{0,1,\dots ,A^2\}$. We prove that, for any real $\tau \ge 2$ and any non-empty proper subset $J\left(b\right)$ of $A$ with at least two elements, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as power series in b with coefficients in $J\left(b\right)$ and with the irrationality exponent (in terms of Gaussian integers) equal to τ. One of the key ingredients in our construction is the ‘Folding Lemma’ applied to Hurwitz continued fractions. This motivates a Hurwitz continued fraction analogue of the well-known Zaremba's conjecture. We prove several results in support of this conjecture.