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.
，其中 
是分數部分，對於幾乎所有實數 
都是等分佈的（即，例外集具有勒貝格 測度零）。例外數字包括正整數、白銀比 
(Finch 2003) 和黃金比 
。上面的圖說明了 
對於 
、
、
和 
的分佈。測度為 1 的集合的候選成員很容易找到，但很難證明。然而，列文明確地構建了這樣一個例子 (Drmota and Tichy 1997)。
，最簡單的此類序列 
的性質已被廣泛研究 (Finch 2003)。前幾項是 0, 1/2, 1/4, 3/8, 1/16, 19/32, 25/64, 11/128, 161/256, 227/512, ... (OEIS A002380 和 A000079; Pillai 1936; Lehmer 1941)，如上圖所示 (Wolfram 2002, pp. 121-122)。例如，
在 
和 ![[1/2,1]](/images/equations/PowerFractionalParts/Inline15.svg)
中都有無限個累積點 (Pisot 1938, Vijayaraghavan 1941)。此外，Flatto等人。(1995) 證明，[0,1] 的任何子區間，如果包含 
的所有累積點，至多除去有限個，則其長度必須至少為 1/3。令人驚訝的是，序列 
也與考拉茲猜想和華林問題有關。
的數字，其中 
是分數部分，出現在華林問題中。特別是，如果滿足不等式，華林問題可以完全解決![frac[(3/2)^n]<=1-(3/4)^n](/images/equations/PowerFractionalParts/NumberedEquation1.svg)
![(3/4)^n<frac[(3/2)^n]<1-(3/4)^n](/images/equations/PowerFractionalParts/NumberedEquation2.svg)
(Bennett 1993, 1994; Finch 2003)。此外，常數 3/4 可以減小到 0.5769 (Beukers 1981, Dubitskas 1990)。遺憾的是，這些不等式尚未得到證明。
." Russian Math. Survey 45, 163-164, 1990.Finch, S. R. "Powers of 3/2 Modulo One." §2.30.1 in 
." 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.