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 $\mathcal A=\{0,1,\ldots, A^2\}$. We prove that, for any real $τ\geq 2$ and any non-empty proper subset $J(b)$ of $\mathcal A$, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as a power series in $b$ with coefficients in $J(b)$ 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.