John Blackman

Restrictions on the base-p expansions of putative counterexamples to the p-adic Littlewood Conjecture

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):=liminf_q 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., liminf_k M(p^{k}x)=0.

In this talk, I will briefly discuss the history of the problem and then move on to discussing some combinatorial conditions that are imposed on the base-p expansion of any counterexample to the p-adic Littlewood Conjecture. This will lead on to the proof that for each fixed d>0 the length of the (2+d)-powers that appear in the base-p expansion of a counterexample to pLC are bounded. I will then show that if a counterexample exists, so does a counterexample whose base-p expansion is uniformly recurrent. Finally, I will discuss the special case when p=2 and show that the conjecture holds for all pure morphic words. This is based on joint work with S. Kristensen and M.J. Northey.