Equidistributed Sequence
A sequence of real numbers 
is equidistributed
on an interval ![[a,b]](/National_Library/im_/https://mathworld.wolfram.com/images/equations/EquidistributedSequence/Inline2.gif)
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]](/National_Library/im_/https://mathworld.wolfram.com/images/equations/EquidistributedSequence/Inline4.gif)
.
However, dense sets need not necessarily be equidistributed. For example, 
, where
is the fractional
part, is dense in ![[0,1]](/National_Library/im_/https://mathworld.wolfram.com/images/equations/EquidistributedSequence/Inline7.gif)
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.
 | Sloane |  with no empty intervals |
 | A036413 | 1, 2, 3, 4, 6, 7, 32, 35, 39, 71, 465, 536, 1001, ... |
 | A046158 | 1, 2, 3, 5, 6, 7, 12, 19, 26, 97, 123, 149, 272, 395, ... |
 | A036415 | 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89, 144, ... |
 | A036417 | 1, 6, 7, 106, 112, 113, 33102, 33215, ... |
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