Some bounds related to the 2-adic Littlewood conjecture

Abstract: Consider alpha = (sqrt(17)-1)/8. One can check that all partial quotients in the continued fraction expansion of alpha are bounded by 3. If we multiply alpha by 2, we get a number where again all partial quotients are bounded by 3. And the same is true for 4*alpha. Might this go on forever as we keep multiplying by 2 (mod 1)? Of course, the answer is “no”, as the 2-adic Littlewood conjecture is known to be true for quadratic irrationals. In this talk, we will use Hurwitz’s algorithm for multiplication by 2 to approach the 2-adic Littlewood conjecture in a completely naive way, and we will (im)prove some bounds related to the 2-adic Littlewood conjecture and a variant of it. Joint work with Dinis Vitorino.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar