One of the main themes of Diophantine approximation is to study how well a real number $x$ can be approximated by a rational number $p/q$ accounting for the size of the denominator $q$. One can measure how well a real number can be rationally approximated by computing the Markov constant $M(x):=\text{lim inf } q^2|x-p/q|$. If the Markov constant is $0$ then $x$ is well approximable. For $M(x)'>0$ the number is badly approximable -- with larger values of $M(x)$ representing worse rates of approximation.

In a similar vein, the $p$-adic Littlewood Conjecture asks if, given a prime $p$ and a badly approximable number $x$, one can always find a subsequence of $p^{k}x$ such that the Markov constant of this sequence tends to $0$, i.e. if $m_p(x):= \text{lim inf } M(p^{k}x)=0$. Whilst this Conjecture has gathered a significant amount of interest, it has proven to be very difficult to solve directly. Instead, one can try and upper bounds for $m_p(x)$ for a fixed prime $p$. This general idea was explored by Badziahin, who produced an algorithm that showed $m_p(x)$ is bounded above by $1/9$ for $p=2$. This algorithm took 3 seconds to compute this bound. However, when the same algorithm tried to compute a bound of $1/10$, it ran for over 60 hours without a conclusive result.

In this talk, I will outline a brief history of the $p$-adic Littlewood Conjecture and discuss how hyperbolic geometry can be used to help understand the problem further. Finally, I will describe an alternative algorithm for obtaining upper bounds on $m_p(x)$ and discuss the results of this algorithm: most notably, that we were able to show that $m_p(x)$ is bounded above by $1/18$ for $p=2$.

This talk is based on joint work with Dr M.J. Northey.