On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle

Abstract

Let the continued fraction expansion of any irrational number $t \in (0,1)$ be denoted by $[0,a_{1}(t),a_{2}(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \[ S=\{t \in (0,1): a_{i+1}(t) \geq \phi^{d_{i}(t)} \text{infinitely often}\}, \] where $\phi = (\sqrt{5}+1)/2$. Let $Y_{S} =\{\exp(2 \pi i t): t \in S \}$. It is shown that if $y \in Y_{S}$ then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an uncountable set of measure zero. It is also shown that there is an uncountable set of points, $G \subset Y_{S}$, such that if $y \in G$, then R(y) does not converge generally. It is further shown that R(y) does not converge generally for |y| > 1 and that R(y) does converge generally if y is a primitive 5m-th root of unity, some $m \in \mathbb{N}$.