In some of my recent research efforts, I've been applying a lot of estimates for exponential sums involving exponent pairs. Two seemingly simple questions have arisen from these calculations, and I present them here as open questions:
Q1. For any positive integer $r$ and any $\epsilon > 0$, does there exist an exponent pair $(k,l)$, producible only from the $A$ and $B$ processes applied to the pair $(0,1)$, such that $$ \left| \frac{l}{k} - r \right| < \epsilon? $$
Q2. Does the set of exponent pairs satisfying this for smaller and smaller $\epsilon$ have a limit point (in the standard topology on $\mathbb{R}^2$)? If so, is this limit point unique?
In the case of $\epsilon = 0$, one can find exponent pairs for some small values of $r$ via a short computer calculation. For instance, in the cases $r=2,3,4,6$: $$ \begin{aligned} &\left(\frac{2}{7},\frac{4}{7}\right) = BA^2\left(\frac{1}{2},\frac{1}{2}\right), \qquad &&\left(\frac{11}{53},\frac{33}{53}\right) = BABA^2BA^2\left(\frac{1}{2},\frac{1}{2}\right), \\ &\left(\frac{1}{6},\frac{2}{3}\right) = A\left(\frac{1}{2},\frac{1}{2}\right), &&\left( \frac{669}{5636},\frac{2007}{2818}\right) =ABA^2BA^2BABABABA^4\left(\frac{1}{2},\frac{1}{2}\right). \end{aligned} $$
Graham and Kolesnik, in their well known book on van Der Corput's method, frame the $A$ and $B$ processes as linear transformations in projective space. Perhaps one can answer these questions using a clever linear algebra argument, but I have not been able to do so. Or, perhaps someone has already answered these questions in the exponent pair literature (though they do seem a bit esoteric).
Any comments, ideas, or references are much appreciated.
