The Littlewood conjecture states that for any two real numbers 
,
 |
where 
denotes the nearest integer function.
In layman's terms, this conjecture concerns the simultaneous approximation of two real
numbers by rationals, indeed saying that any
two real numbers 
and 
can be simultaneously approximated at least moderately well
by rationals having the same denominator
(Venkatesh 2008).
Though a proof of the Littlewood conjecture remains an open problem, many partial results exist. The set of exceptional
pairs 
of real numbers 
and 
for which the conjecture fails
has Lebesgue measure zero.
Einsiedler et al. (2006) proved the stronger result that this set
has Hausdorff dimension zero.
