Abstract
This article is an exposition of the main result of [Hoc12], that self-similar sets whose dimension is smaller than the trivial upper bound have “almost overlaps” between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in [Hoc12], but also falls short of a complete proof; in the last section we discuss how the heuristic argument is turned into a rigorous one.
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- 1.
The mddle-1/3 Cantor set can also be described in other ways, e.g. by a recursive construction, or symbolically as the points in \([0,1]\) that can be written in base \(3\) without the digit \(1\). General self-similar sets also have representations of this kind, but in this paper we shall not use them.
- 2.
Iterated function systems consisting of non-affine maps and on other metric spaces than \(\mathbb {R}\) are also of interest, but we do not discuss them here.
- 3.
Supported by ERC grant 306494
- 4.
It would be better to write \(\text {sdim}\Phi \), since this quantity depends on the presentation of \(X\) and not on \(X\) itself, but generally there is only one IFS given and no confusion should arise.
- 5.
This notion is identical to a flow on the tree in the sense of network theory.
- 6.
In general there is a similar formula: \(\mu *\nu =\int \int \delta _{x+y}d\mu (x)d\nu (y)\), where the integral is interpreted as a measure by integrating against Borel functions.
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Many thanks to Boris Solomyak for his coments on the paper.
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Hochman, M. (2014). Self Similar Sets, Entropy and Additive Combinatorics. In: Feng, DJ., Lau, KS. (eds) Geometry and Analysis of Fractals. Springer Proceedings in Mathematics & Statistics, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43920-3_8
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DOI: https://doi.org/10.1007/978-3-662-43920-3_8
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