Sampling to Characterize Cantor Sets Sarah McCarty University of - PowerPoint PPT Presentation

Sampling to Characterize Cantor Sets Sarah McCarty University of Nebraska at Omaha Allison Byars, Evan Camrud, Nate Harding, Keith Sullivan, Eric Weber Iowa State University February 2, 2020

Outline 1 Background 2 The Problem 3 Tools 4 Using Samples

Cantor Sets: Scaling Factor N , Digit Set D , Vector B

Cumulative Distribution Function (CDF) Figure: Cantor Set (1,0,1) and its CDF

Outline 1 Background 2 The Problem 3 Tools 4 Using Samples

Goal Goal: Knowing F is the CDF of a Cantor Set with scaling factor no more than K , choose sample points to determine F Figuring out N and D of the Cantor set.

Intersections are the Problem

Outline 1 Background 2 The Problem 3 Tools 4 Using Samples

Definition Definition Define r and s to be multiplicatively dependent , denoted r ∼ s , if ∃ a, b ∈ N such that r a = s b . Example: 9 and 27 are multiplicatively dependent as 9 3 = 27 2 Example: 3 and 6 are not mutliplicatively dependent We divide into two cases: multiplicatively dependent and independent scaling factors.

Multiplicatively Dependent

Multiplicatively Independent

Multiplicatively Dependent Scaling Factors Let ⊗ be the Kronecker product. Let B L have scaling factor Z L and B M have scaling factor Z M . Lemma If B L ⊗ B M = B M ⊗ B L , then F B L = F B M . Theorem Z L + M } Z L + M − 1 m S = { is sufficient to differentiate F B L and F B M . m =1

Multiplicatively Independent: Rationality and the CDF Lemma Let C N,D be a Cantor set and F N,D the CDF. For x ∈ Q c ∩ [0 , 1] , F N,D ( x ) ∈ Q c if and only if x ∈ C N,D .

Multiplicatively Independent: Normal Theorem Definition Number normal to base N : all finite sequences of digits appear equally often in the N -ary expansion Numbers normal to base N are never in Cantor sets with scaling factor N Theorem For x ∈ C N,D , almost all x are normal to every base M > 1 such that M �∼ N .

Outline 1 Background 2 The Problem 3 Tools 4 Using Samples

Sampling First, we eliminate all Cantor sets with multiplicatively independent scaling factors: Multiplicatively Independent points ( O ( K 3 )) For all possible M ≤ K and all possible digits sets D i,M such that | D i,M | = 2, pick an irrational x i,M normal to all bases �∼ M When M �∼ N , F ( x i,M ) ∈ Q for all x i,M When M = N , at least one i , F ( x i,N ) ∈ Q c Building digit sets

Using Samples Then, we are guaranteed that all remaining possibilities can be differentiated with this set: Multiplicatively Dependent points ( O ( K 3 )) � m � M − 1 M m =1 for all M ∈ { 2 2 , 3 2 , ..., K 2 }

Conclusion Thus, we are able to determine N and D with O ( K 3 ) points

References [1] Broxson, B. (2006). The Kronecker product. UNF Graduate Theses and Dissertations . 25. [2] Cassels, J. W. S. (1959). On a problem of Steinhaus about normal numbers. Colloquium Mathematicae, 7 (1). [3] Hutchinson, J. E. (1981). Fractals and self-similarity. Indiana Univ. Math. J., 5 (30). [4] Pollington, A. D. (1988). The Hausdorff dimension of a set of normal numbers. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 44 (2). [5] Schmidt, W. M. (1962). ¨ Uber die Normalitt von Zahlen zu verschiedenen Basen. Acta Arithmetica, 7 (3). [6] Weyl, H. (1916). ¨ Uber die Gleichverteilung von Zahlen mod Eins. Mathematische Annalen , 77.

Thank you!

Outline 1 Background 2 The Problem 3 Tools 4 Using Samples