DISTRIBUTION OF DIGITS IN INTEGERS: BESICOVITCH–EGGLESTON SUBSETS OF ${\mathbb N}$

Abstract

Fix an integer N ⩾ 2. For a positive integer n ∈ N, let n = d0(n) + d1(n)N + d2(n)N2+…+dγ(n)(n)Nγ(n) where di(n) ∈ {0,1,2,…,N − 1} and dγ(n)(n) ≠ 0 denote the N-ary expansion of n. For a probability vector p = (p0…,pN−1) and r > 0, the r approximative discrete Besicovitch–Eggleston set Br(p) is defined by B r ( p ) = { ∈ N | | | { 0 ⩽ k ⩽ γ ( n ) | d k ( n ) = i } | γ ( n ) + 1 − p i | ⩽ r forall i } , that is, Br(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 pi by less than r for all i∈ {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 δ(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 Br(p) are computed (in the limit as rs 0). Let p=(p0,…,pN−1) be a probability vector. Then lim r → 0 dim ⇀ ( B r ( p ) ) = lim r → 0 dim ¯ ( B r ( p ) ) = lim r → 0 δ ( B r ( p ) ) = − ∑ 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.