Hardy and Littlewood (1914) proved that the sequence , where is the fractional part,
is equidistributed for almost all
real numbers
(i.e., the exceptional set has Lebesguemeasure
zero). Exceptional numbers include the positive integers, the silver
ratio
(Finch 2003), and the golden ratio . The plots above illustrate the distribution of for , , , and . Candidate members of the measure one set are
easy to find, but difficult to prove. However, Levin has explicitly constructed such
an example (Drmota and Tichy 1997).

The properties of , the simplest such sequence for a rational number
,
have been extensively studied (Finch 2003). The first few terms are 0, 1/2, 1/4,
3/8, 1/16, 19/32, 25/64, 11/128, 161/256, 227/512, ... (OEIS A002380
and A000079; Pillai 1936; Lehmer 1941), plotted
above (Wolfram 2002, pp. 121-122).
For example,
has infinitely many accumulation points in
both
and
(Pisot 1938, Vijayaraghavan 1941). Furthermore, Flatto et al. (1995) proved
that any subinterval of containing all but at most finitely many accumulation
points of
must have length at least 1/3. Surprisingly, the sequence is also connected with the Collatz
problem and with Waring's problem.

holds. No counterexample to this inequality is known, and it is even believed that it can be extended to

for
(Bennett 1993, 1994; Finch 2003). Furthermore, the constant 3/4 can be decreased
to 0.5769 (Beukers 1981, Dubitskas 1990). Unfortunately, these inequalities have
not been proved.

Bennett, M. A. "Fractional Parts of Powers of Rational Numbers." Math. Proc. Cambridge Philos. Soc.114, 191-201,
1993.Bennett, M. A. "An Ideal Waring Problem with Restricted
Summands." Acta Arith.66, 125-132, 1994.Beukers,
F. "Fractional Parts of Powers of Rational Numbers." Math. Proc. Cambridge
Philos. Soc.90, 13-20, 1981.Drmota, M. and Tichy, R. F.
Sequences,
Discrepancies and Applications. New York: Springer-Verlag, 1997.Dubitskas,
A. K. "A Lower Bound for the Quantity ." Russian Math. Survey45, 163-164,
1990.Finch, S. R. "Powers of 3/2 Modulo One." §2.30.1
in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 194-199,
2003.Flatto, L.; Lagarias, J. C.; Pollington, A. D. "On
the Range of Fractional Parts ." Acta Arith.70, 125-147, 1995.Hardy,
G. H. and Littlewood, J. E. "Some Problems of Diophantine Approximation."
Acta Math.37, 193-239, 1914.Lehmer, D. H. Guide
to Tables in the Theory of Numbers. Bulletin No. 105. Washington, DC:
National Research Council, p. 82, 1941.Pillai, S. S. "On
Waring's Problem." J. Indian Math. Soc.2, 16-44, 1936.Pisot,
C. "La répartition modulo 1 et les nombres algébriques."
Annali di Pisa7, 205-248, 1938.Sloane, N. J. A.
Sequences A000079/M1129 and A002380/M2235
in "The On-Line Encyclopedia of Integer Sequences."Vijayaraghavan,
T. "On the Fractional Parts of the Powers of a Number (I)." J. London
Math. Soc.15, 159-160, 1940.Vijayaraghavan, T. "On
the Fractional Parts of the Powers of a Number (II)." Proc. Cambridge Phil.
Soc.37, 349-357, 1941.Vijayaraghavan, T. "On the Fractional
Parts of the Powers of a Number (III)." J. London Math. Soc.17,
137-138, 1942.Wolfram, S. A
New Kind of Science. Champaign, IL: Wolfram Media, pp. 121-122,
2002.