On Minkowski’s singular function

Abstract

The function f on the closed unit interval I to be considered here was introduced by Minkowski. It maps the algebraic irrationalities of degree at most 2 continuously to the rationals in I. If z = a/b and z' = c/d are (reduced) fractions, then we write z * z' for the mediant (a + c)/(b + d) and z z' for the arithmetic mean. Let Q denote the (dense) set of rationals in I with terminating dyadic expansion, i.e. the set of numbers x= n2'' (n = 0,.. .,2-1, v E N). Define a function g on Q inductively as follows: Let y' = 0, y= 1. Suppose the values yn = g(Xn) have been defined on level v. Then put y2+' = Yk (0 < k < 2W) and insert +1 =YV1 *Yk (1< k <2W) to obtain the images y+' = g(x+') on level v+ 1. The resulting function g is a strictly increasing continuous bijection from Q to the rationals. Note that g is constructed so as to have the property g(x) *g(x') = g(x x') whenever x, x' E Q, x' x = 21-, v E N. The inverse of g can be extended to the reals to yield a strictly increasing continuous distribution function f inducing a (singular) probability measure on I. We will be concerned with the regular and semiregular continued fraction expansions y=J +l 3+ [= la),...],l ,,... E N and y = a-i b= [a, b, ... .1, a, b, ... E N {1}. In the dyadic expansion of elements x E I we collect 0's and l's and write x = 0, -110O1 60, , letting a 1,/,-y,6, ,. . . denote string lengths (a, 3, -y, 6, E, ... E N). The following has been proved by different arguments (see [2] for references): (i) a+1;2i_j,'-y+2,26_1,E+2, ... f'(011 ,0,_1 16 0, ) (this is [2, (5)]); (ii) f is singular. Our purpose here is to present a simple unified approach to both properties by splitting up (i) into the following equalities, the second one making (ii) evident: