Equidistributed Sequence

A sequence of real numbers  is equidistributed on an interval  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 .

However, dense sets need not necessarily be equidistributed. For example, , where  is the fractional part, is dense in  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, ...

REFERENCES:

Hardy, G. H. and Littlewood, J. E. "Some Problems of Diophantine Approximation." Acta Math. 37, 193-239, 1914.

Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, 1974.

Pólya, G. and Szegö, G. Problems and Theorems in Analysis I. New York: Springer-Verlag, p. 88, 1972.

Sloane, N. J. A. Sequences A036412, A036413, A036414, A036415, A036416, A036417, A046157, and A046158 in "The On-Line Encyclopedia of Integer Sequences."

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 155-156, 1991.