Minkowski-Bower Constant

The Minkowski-Bower constant is the smallest positive fixed point of Minkowski's question mark function (Bower 1999, Finch 2003, pp. 441-443). Its value is

(1)

(OEIS A048819). The continued fractions of and are

(2)

(OEIS A058914) and

(3)

(OEIS A120221), respectively.

The symmetry for implies that is also a fixed point. It is conjectured that 0, 1/2, , , and 1 are the only fixed points on , giving four modulo 1 (Gayfulin and Shulga 2020).

Gayfulin and Shulga (2020) proved that the smallest and greatest fixed points in both have irrationality measure 2, so in particular this holds for . For either such fixed point , its partial denominators satisfy and

(4)

for every positive integer . These results do not assume the uniqueness conjecture.