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 2007).
Though proof of the Littlewood conjecture still remains an open problem, many partial results exist. For example, Borel showed that the set of exceptional pairs 
of real
numbers 
and 
for which the conjecture fails has Lebesgue measure zero. Much later, Einsiedler et al. (2006) proved
that the set of pairs of exceptional points also has Hausdorff
dimension zero.
