2023/10/18 by Gerardo González Robert, Robert, Gerardo González, Mumtaz Hussain +3 · 2 citations
Mathematics · #Mathematical Dynamics and Fractals #Analytic Number Theory Research #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.2310.11698
Given b=-A± 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,…, A2\. We prove that, for any real τ≥ 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.