How do we prove the similarity dimension equals the Hausdorff dimension if the self-similar set satisfies the open set condition? Which article contains this proof?
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The open set condition, and it use to establish Hausdorff dimension, is due to
Moran, P. A. P., Additive functions of intervals and Hausdorff measure, Proc. Camb. Philos. Soc. 42, 15-23 (1946). ZBL0063.04088.
Moran was a student of Besicovitch at Cambridge at the time.
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Section 6.5 in
Edgar, Gerald, Measure, topology, and fractal geometry, Undergraduate Texts in Mathematics, 2nd ed. New York, NY: Springer (ISBN 978-0-387-74748-4/hbk). xv, 268 p. (2008). ZBL1152.28008.
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2$\begingroup$ +1 Can I also suggest Hutchinson's 1981 paper as a pretty approachable introduction to self-similar fractal sets? That tends to be the one that I recommend to clever undergrads as a good introduction. And, as long as I am here, thank you for your book, Dr Edgar. It was one of my first tastes of fractal geometry. (Though I miss the first edition, in which all of the proofs ended with happy little smiley faces. What happened to those?) $\endgroup$– Xander Henderson ♦Commented Jun 17, 2019 at 15:37