The God Function G(x,n,j)?

Question

Let the function G(x,n,j) equal 1 when there is a sequence of integers x of length n contained in some real number j and zero when this is not true. (For example, G(9,2,pi)=1 because there is a sequence of 2 9s at about the hundreth decimal place.)

Let the function G(x,n,j) be given in the form of a triple infinte series:

G(x,n,j) = lim ∑ ∑ ∑a(x,q) + b(n,w) + c(j,e),
summation →∞

where the varibles q, w and e are indicies of summation. Let the functions a, b and c be continuous everywhere.

(a) Is it true that

lim ∂G/∂j = k, k < ∞ ?
j→∞

(b) Is it true that for intervals (0, j) , (j+1, j+2), etc, that the following inequalities are always true?
j
∫G(x,j,n) dj <
0

2j
∫G(x,j,n) dj < ... n→∞
j

(For part b assume that if a function f(x) has an set of values for which the Riemann integral fails to exist than the Riemann integral is zero at those values.)

Answer 1

This is an incredibly interesting problem. Although I do not at present have the solution, I will spend some today playing with it and repost anything I discover which I feel will be helpful to you.