Littlewoods conjecture

by Andrew Pollington

Institution: Director - Algebra, Number Theory and Combinatorics panel, National Science Foundation
Date: Thu 3rd September 2009
Time: 1:00 PM
Location:

Abstract: Littlewood's conjecture states that for any pair of real numbers $\alpha$ and $\beta$ then

$$inf_{q\in Z^+}q\|q\alpha\| \|q\beta\| =0.$$

This conjecture remains open but considerable progress has been made recently.

The conjecture is trivially true if one of the numbers is not badly approximable. So we may restrict our attention to badly approximable pairs. It is also the case that if $i+j=1$ with $0\le i\le 1$ we say that $(\alpha,\beta)\in Bad(i,j)$ if there is a constant $c>0$, depending only on $\alpha$ and $\beta$, so that

$$max(\|q\alpha\|^{1/i},\|q\beta\|^{1/j} >c/q,$$ for all natural numbers $q$.

It is clear that any counter example to Littlewood's conjecture must be in $Bad(i,j)$ for any choice of $i$. So if we could show that there are no pairs in the intersection say of Bad(1/3,2/3) and Bad(2/3,1/3) then we would have proved Littlewoods conjecture. Schmidt's conjecture is that this intersection is not empty. We show here that this conjecture is true and so one cannot prove Littlewood's conjecture this way.

This is joint work with Dzmitry Badziahin and Sanju Velani, both at York.