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Sampling to Characterize Cantor Sets Sarah McCarty University of - - PowerPoint PPT Presentation
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
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Cantor Sets: Scaling Factor N, Digit Set D, Vector B
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Cumulative Distribution Function (CDF)
Figure: Cantor Set (1,0,1) and its CDF
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Outline
1 Background 2 The Problem 3 Tools 4 Using Samples
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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.
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Intersections are the Problem
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Outline
1 Background 2 The Problem 3 Tools 4 Using Samples
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Definition
Definition Define r and s to be multiplicatively dependent, denoted r ∼ s, if ∃ a, b ∈ N such that ra = sb. Example: 9 and 27 are multiplicatively dependent as 93 = 272 Example: 3 and 6 are not mutliplicatively dependent We divide into two cases: multiplicatively dependent and independent scaling factors.
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Multiplicatively Dependent
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Multiplicatively Independent
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Multiplicatively Dependent Scaling Factors
Let ⊗ be the Kronecker product. Let BL have scaling factor ZL and BM have scaling factor ZM. Lemma If BL ⊗ BM = BM ⊗ BL, then FBL = FBM . Theorem S = {
m ZL+M }ZL+M−1 m=1
is sufficient to differentiate FBL and FBM .
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Multiplicatively Independent: Rationality and the CDF
Lemma Let CN,D be a Cantor set and FN,D the CDF. For x ∈ Qc ∩ [0, 1], FN,D(x) ∈ Qc if and only if x ∈ CN,D.
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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 ∈ CN,D, almost all x are normal to every base M > 1 such that M ∼ N.
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Outline
1 Background 2 The Problem 3 Tools 4 Using Samples
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Sampling
First, we eliminate all Cantor sets with multiplicatively independent scaling factors: Multiplicatively Independent points (O(K3)) For all possible M ≤ K and all possible digits sets Di,M such that |Di,M| = 2, pick an irrational xi,M normal to all bases ∼ M When M ∼ N, F(xi,M) ∈ Q for all xi,M When M = N, at least one i, F(xi,N) ∈ Qc Building digit sets
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Using Samples
Then, we are guaranteed that all remaining possibilities can be differentiated with this set: Multiplicatively Dependent points (O(K3)) m M M−1
m=1
for all M ∈ {22, 32, ..., K2}
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Conclusion
Thus, we are able to determine N and D with O(K3) points
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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
- f 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.
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Thank you!
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