The Erdős-Littlewood-Offord inequality states that if 
, ..., 
are real numbers satisfying
,
then at most the binomial coefficient 
of the 
signed sums 
, where each 
, lie in any open
interval of length 2 (Erdős 1945).
Equivalently, if 
, ..., 
are independent
and identically distributed random signs, each taking the values 
and 1 with equal probability,
then for every open interval 
of length 2,
 |
The inequality is sharp, as can be seen by taking 
and choosing an open
interval containing a mode of the signed sum.
