A sequence of real numbers 
is equidistributed on an interval ![[a,b]](/images/equations/EquidistributedSequence/Inline2.svg)
if the probability of finding 
in any subinterval is proportional to the subinterval length.
The points of an equidistributed sequence form a dense
set on the interval ![[a,b]](/images/equations/EquidistributedSequence/Inline4.svg)
.
However, dense sets need not necessarily be equidistributed. For example, 
, where 
is the fractional part,
is dense in ![[0,1]](/images/equations/EquidistributedSequence/Inline7.svg)
but not equidistributed, as illustrated above for 
to 5000 (left) and 
to 
(right)
Hardy and Littlewood (1914) proved that the sequence 
, of power
fractional parts is equidistributed for almost all real numbers 
(i.e., the exceptional set has Lebesgue measure zero). Exceptional numbers include the positive
integers, the silver ratio 
(Finch 2003), and the golden
ratio 
.
The top set of above plots show the values of 
for 
equal to e, the Euler-Mascheroni
constant 
,
the golden ratio 
, and pi. Similarly, the bottom set
of above plots show a histogram of the distribution of 
for these constants. Note that while
most settle down to a uniform-appearing distribution, 
curiously appears nonuniform after 
iterations. Steinhaus (1999) remarks that the highly uniform
distribution of 
has its roots in the form of the continued
fraction for 
.
Now consider the number of empty intervals in the distribution of 
in the intervals bounded by the intervals
determined by 0, 
, 
, ..., 
, 1 for 
, 2, ..., summarized below for the constants previously considered.
 | Sloane | # empty intervals
for  ,
2, ... |
 | A036412 | 0, 0, 0, 0, 1, 0, 0, 1, 1, 3, 1, 4, 4, 7, 5, ... |
 | A046157 | 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 3, 0, 3, 5, 3, ... |
 | A036414 | 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, ... |
 | A036416 | 0, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 4, 5, 7, 7, ... |
The values of 
for which no bins are left blank are given in the following table.
with no empty intervals
