Normality preserving operations for Cantor series expansions and associated fractals, I

Abstract

We investigate how non-zero rational multiplication and rational addition affect normality with respect to Q-Cantor series expansions. In particular, we show that there exists a Q such that the set of real numbers which are Q-normal but not Q-distribution normal, and which still have this property when multiplied and added by rational numbers has full Hausdorff dimension. Moreover, we give such a number that is explicit in the sense that it is computable.

Non-trivial matrix actions preserve normality for continued fractions

“…We then define all the S j recursively, deriving S j by letting it be the coarsest refinement of S j−1 that includes all cylinder sets C s with the sum of digits in s equal to j. Thus S 1 = {Ω \ C [1] , C [1] }, S 2 = {Ω \ (C [1] ∪ C [2] ), C [2] , C [1] \ C [1,1] , C [1,1] }, and so on. Note that S j always consists of just cylinders and anti-cylinders.…”

“…Namely, we remove digits from the head or tail of s and append them to s + or prepend tem to s − until removing any more digits would cause r to no longer appear nicely within t at the same relative position. We refer to this (s, M ) with the length of s minimized as a trigger string for r. If there are exactly k copies of r in t all of which have the same minimal decomposition, we say that (s, M ) is a trigger string of multiplicity k. For example if the resultant string of ( [10], M ) was [1; 1, 1, 1, 1, 1] and the desired string r = [1], then this string has multiplicity 2.…”

“…Email: vandehey@uga.edu. 1 For readability's sake, we will no longer say "to base 10" every time. The results mentioned here would follow if we replaced 10 by any integer base b ≥ 2.…”

“…2 Simple normality asks that all of the one-digit strings appear with the correct limiting frequency. 1 non-zero rational multiplication preserved CF-normality [6,Problem 10.56,pp. 222].…”

“…. , c k ], we define the cylinder set of s to be C s = {x ∈ [0, 1) : a i (x) = c i , 1 ≤ i ≤ k}, and we say this cylinder set has rank k. We shall also need the usual matrix action on real numbers given by α β γ δ x = αx + β γx + δ With these definitions in mind, we say that a point x ∈ [0, 1) is CF-normal if for any string s, we have (1) lim n→∞ #{0 ≤ i < n : T i x ∈ C s } n = µ(C s ).…”