Distribution of digits in integers: Besicovitch-Eggleston subsets of N

Fix an integer N greater than or equal to 2. For a positive integer n is an element of N, let n = d(0)(n) + d(1)(n)N + d(2)(n)N-2 + (...) + d(gamma(n))(n)N-gamma(n) where d(i)(n) is an element of {0,1,2,...,N-1} and d(gamma(n))(n) not equal 0 denote the N-ary expansion of n. For a probability vector p = (p(0),...,p(N-1)) and r > 0, the r approximative discrete Besicovitch-Eggleston set B-r(P) is defined

B-r(p) = {n is an element of N\\\{0 less than or equal to k less than or equal to gamma(n) \ d(k)(n) = i}\/gamma(n) + 1 - p(i)\ less than or equal to r for all i},

that is, B-r(P) is the set of positive integers n such that the frequency of the digit i in the N-ary expansion of n differs from p(i) by less than r for all i is an element of {0,1,2,...,N - 1}. Three natural fractional dimensions of subsets E of N are defined, namely, the lower fractional dimension dim(E), the upper fractional dimension dim(E) and the exponent of convergence delta(E), and the dimensions of various subsets of N defined in terms of the frequencies of the digits in the N-ary expansion of the positive integers are studied. In particular, the dimensions of B-r(p) are computed (in the limit as r SE arrow 0). Let p = (p(0),...,p(N-1)) be a probability vector. Then

lim(rSE arrow0) dim(B-r(p)) = lim(rSE arrow0) dim(B-r(p)) = lim(rSE arrow0) delta(B-r(p)) = -Sigma(i) p(i) log p(i)/log N.

This result provides a natural discrete analogue of a classical result due to Besicovitch and Eggleston on the Hausdorff dimension of certain sets of non-normal numbers. Several applications to the theory of normal numbers are given.