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Average distances between points in graph-directed self-similar fractals

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Olsen_2018_MN_Averagedistances_AAM.pdf (661.0Kb)
Date
27/08/2018
Author
Olsen, L.
Richardson, A.
Keywords
Average distance
Drobot–Turner set
Graph-directed self-similar measures
Graph-directed self-similar sets
Hausdorff measure
28A78
QA Mathematics
T-NDAS
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Abstract
We study several distinct notions of average distances between points belonging to graph‐directed self‐similar subsets of ℝ. In particular, we compute the average distance with respect to graph‐directed self‐similar measures, and with respect to the normalised Hausdorff measure. As an application of our main results, we compute the average distance between two points belonging to the Drobot–Turner set TN(c,m) with respect to the normalised Hausdorff measure, i.e. we compute 1/Hs(TN(c,m)2 ∫TN(c,m)2|x-y|(Hs x Hs(x,y) where s denotes the Hausdorff dimension of is the s‐dimensional Hausdorff measure; here the Drobot–Turner set (introduced by Drobot & Turner in 1989) is defined as follows, namely, for positive integers N and m and a positive real number c, the Drobot–Turner set TN(c,m) is the set of those real numbers x ε [0,1] for which any m consecutive base N digits in the N‐ary expansion of x sum up to at least c. For example, if N=2, m=3 and c=2, then our results show that  1/Hs(T2(2,3))2 ∫T2(2,3)2 |x-y|d(Hs x Hs)(x,y) = 4444λ2 + 2071λ + 3030 / 1241λ2 + 5650λ + 8281 = 0.36610656 ..., where λ = 1.465571232 ... is the unique positive real number such that λ3 - λ2 - 1 = 0.
Citation
Olsen , L & Richardson , A 2018 , ' Average distances between points in graph-directed self-similar fractals ' , Mathematische Nachrichten , vol. Early View . https://doi.org/10.1002/mana.201600354
Publication
Mathematische Nachrichten
Status
Peer reviewed
DOI
https://doi.org/10.1002/mana.201600354
ISSN
0025-584X
Type
Journal article
Rights
© 2018, WILEY-VCH Verlag GmbH & Co KGGaA, Weinheim. This work has been made available online in accordance with the publisher’s policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1002/mana.201600354
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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/18376

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