The Kempner number^[1] is the sum of the series

${\displaystyle \kappa$ :=2^{-2^{0}}+2^{-2^{1}}+2^{-2^{2}}+\cdots =\sum _{n\geq 0}2^{-2^{n}}.}

It is named after Aubrey Kempner, who proved it transcendental in 1916.^[2] It is an example of a number easy to prove transcendental which is not a Liouville number.^[1]: §1

Properties

By definition, the binary expansion of the Kempner number has zeroes everywhere except at places which are powers of two:

κ = 0.110100010000000100000000000000010000000000000000000000000000000100... (base two.)

Since the first proof of transcendence by Kempner, many other proofs have been given; see the references.^{[1][3][4][5][6][7][8][9]}

Jeffrey Shallit has proven that it has a simple continued fraction expansion, obtainable by the following construction:^{[10]: Theorem 1}

1. Start with the partial expansion [0, 1, 3].
2. If the partial expansion is [a, b, ..., y, z], replace it by [a, b, ..., y, z + 1, z − 1, y, ..., b].
3. If this generated a zero, replace [..., a, 0, b, ...] by [..., a + b, ...].
4. Repeat steps 2 and 3 indefinitely.

This generates the expansion (sequence A007400 in the OEIS)

${\displaystyle [0;1,4,2,4,4,6,4,2,4,6,...]=0+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{2+{\cfrac {1}{4+\ _{\ddots }}}}}}}}}.}$

After the first partial quotients, the remainders are all 2, 4 or 6. Since this continued fraction has bounded partial quotients, the Kempner number has irrationality measure 2.

References