Distribution of Digits in Integers: Besicovitch–Eggleston Subsets of N

Fix an integer N ⩾ 2. For a positive integer n ∈ N , let n = d 0 ( n ) + d 1 ( n ) N + d 2 ( n ) N 2 +…+ d γ( n ) ( n ) N γ ( n ) where d i ( n ) ∈ {0,1,2,…, N − 1} and d γ( n ) ( n ) ≠ 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 byB r ( p ) = { ∈ N | || { 0 ⩽ k ⩽ γ ( n ) | d k ( n ) = i } |γ ( n ) + 1 − p i | ⩽ r forall 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 ∈ {0,1,2,…, N − 1}. Three natural fractional dimensions of subsets E of N are defined, namely, the lower fractional dimensiondim ¯ ( E ) , the upper fractional dimensiondim ¯ ( 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 B r ( p ) are computed (in the limit as r s 0). Let p =( p 0 ,…, p N −1 ) be a probability vector. Thenlim r → 0dim ⇀ ( B r ( p ) ) = lim r → 0dim ¯ ( B r ( p ) ) = lim r → 0 δ ( B r ( p ) ) = −∑ i p i log p ilog NThis 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.