Cantor Function


The Cantor function F(x) is the continuous but not absolutely continuous function on [0,1] which may be defined as follows. First, express x in ternary. If the resulting ternary digit string contains the digit 1, replace every ternary digit following the 1 by a 0. Next, replace all 2's with 1's. Finally, interpret the result as a binary number which then gives F(x).

The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function F_q for q>2, with q=3 corresponding to the usual Cantor function (Gorin and Kukushkin 2004).

Chalice (1991) showed that any real-valued function F(x) on [0,1] which is monotone increasing and satisfies

1. F(0)=0,

2. F(x/3)=F(x)/2,

3. F(1-x)=1-F(x)

is the Cantor function (Chalice 1991; Wagon 2000, p. 132).

Gorin and Kukushkin (2004) give the remarkable identity

 =1/(n+1)-(q-2)sum_(k=1)^(|_n/2_|)(n; 2k)(2^(2k-1)-1)/(q·2^(2k-1)-1)(B_(2k))/(n-2k+1)

for integer n. For q=3 and n=1, 2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, ... (OEIS A095844 and A095845).

M. Trott (pers. comm., June 8, 2004) has noted that

 int_0^1[F(t)]^(F(t))dt approx 0.750387...

(OEIS A113223), which seems to be just slightly greater than 3/4.

See also

Cantor Set, Devil's Staircase

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Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, p. 237, 2007.Chalice, D. R. "A Characterization of the Cantor Function." Amer. Math. Monthly 98, 255-258, 1991.Devaney, R. L. An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, 1987.Gorin, E. A. and Kukushkin, B. N. "Integrals Related to the Cantor Function." St. Petersburg Math. J. 15, 449-468, 2004.Sloane, N. J. A. Sequences A095844, A095845, A113223 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. "The Cantor Function" and "Complex Cantor Sets." §5.2 and 5.3 in Mathematica in Action, 2nd ed. New York: W. H. Freeman, pp. 132-138, 2000.

Referenced on Wolfram|Alpha

Cantor Function

Cite this as:

Weisstein, Eric W. "Cantor Function." From MathWorld--A Wolfram Web Resource.

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