Abstract
This paper is an exposition, with some new applications, of our results from Hochman (Ann Math (2) 180(2):773–822, 2014; preprint, 2015, http://arxiv.org/abs/1503.09043) on the growth of entropy of convolutions. We explain the main result on \(\mathbb{R}\), and derive, via a linearization argument, an analogous result for the action of the affine group on \(\mathbb{R}\). We also develop versions of the results for entropy dimension and Hausdorff dimension. The method is applied to two problems on the border of fractal geometry and additive combinatorics. First, we consider attractors X of compact families \(\Phi\) of similarities of \(\mathbb{R}\). We conjecture that if \(\Phi\) is uncountable and X is not a singleton (equivalently, \(\Phi\) is not contained in a 1-parameter semigroup) then dimX = 1. We show that this would follow from the classical overlaps conjecture for self-similar sets, and unconditionally we show that if X is not a point and \(\dim \Phi> 0\) then dimX = 1. Second, we study a problem due to Shmerkin and Keleti, who have asked how small a set \(\emptyset \neq Y \subseteq \mathbb{R}\) can be if at every point it contains a scaled copy of the middle-third Cantor set K. Such a set must have dimension at least dimK and we show that its dimension is at least dimK + δ for some constant δ > 0.
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Notes
- 1.
If we work in a complete metric space and general contractions, taking the topology of uniform convergence of maps on compact sets, then existence and uniqueness of the attractor is still true if we assume that all \(\varphi \in \Phi\) contract by at least some 0 < r < 1. Without uniformity, existence can fail already for a single map.
- 2.
We use another parameterizations in Sect. 7, but only there.
- 3.
We require that \(\omega \rightarrow \mu ^{\omega }\in \mathcal{ P}(X)\) be measurable in the sense that ω ↦ ∫fdμ ω is measurable for all bounded measurable \(f: X \rightarrow \mathbb{R}\), and the expectation \(\mathbb{E}(\mu )\) is understood the probability measure ν determined by \(\nu (A) = \mathbb{E}(\mu (A))\) for all measurable A, or equivalently, \(\int fd\nu = \mathbb{E}(\int fd\mu )\) for bounded measurable f.
- 4.
Entropy porosity in the sense above is weaker than porosity, since it allows the measure to be fully supported on small balls (i.e., there do not need to be holds in its support). But the upper bound on the entropy of components means that most components are far away from being uniform at a slightly finer scale.
- 5.
Differentiability would be enough for most purposes, but then the error term in (16) would be merely o( | x − x 0 | + | y − y 0 | ) instead of the quadratic error, and later on we will want the quadratic rate.
- 6.
One can also show that μ is exact-dimensional, but we do not need this fact here.
- 7.
To derive this from the sampling theorem for real-valued random variables, note that we need to show that \(\int f\,d\mathbb{E}(\mu _{\tau }) =\int fd\mu\) for all bounded functions f, and this follows since ξ n = ∫fdμ n is easily seen to be a martingale for \((\mathcal{F}_{n})\), and by Fubini \(\int f\,d\mathbb{E}(\mu _{\tau }) = \mathbb{E}(\int fd\mu _{\tau }) = \mathbb{E}(\xi _{\tau }) = \mathbb{E}(\xi _{0}) =\int fd\mu\), where we used the real-valued optional stopping theorem in the second to last equality.
- 8.
One way to see this is by adapting the proof of the classical lemma of Fekete. Alternatively consider \(b_{n} = a_{n} -\sqrt{n}\), which after dividing by n has the same asymptotics as a n , but satisfies b m+n ≥ b m + b n for all m, n large enough, so that Fekete’s lemma applies to it.
- 9.
Here is a proof sketch: Let (X i ) be a martingale with \(\mathbb{E}X_{i} = 0\), \(\mathbb{E}(X_{i}^{2}) \leq a\), and X i ≥ −b for some constants a, b > 0. Let w k, i be as before, write S k = ∑ i = 1 k w k, i X i . We claim that \(\liminf _{k}S_{k} \geq 0\) a.s. Consider first the subsequence \(S_{k^{2}}\). Using w k, i = (1 + τ + o(1))k −(1+τ) i τ and \(\mathbb{E}(X_{i}X_{j}) = 0\) for i ≠ j, we have
$$\displaystyle{\mathbb{E}((S_{k^{2}})^{2}) =\sum _{ i=1}^{k^{2}}w_{ k^{2},i}^{2}\mathbb{E}(X_{ i}^{2}) = O(k^{-4(1+\tau )}\sum _{ i=1}^{k^{2}}i^{2\tau }) = O(k^{-2})}$$Hence by Markov’s inequality \(\sum \mathbb{P}(S_{k^{2}}>\varepsilon ) <\infty\), and by Borel-Cantelli, \(S_{k^{2}} \rightarrow 0\) a.s. We now interpolate: for k 2 ≤ ℓ < (k + 1)2 and using \(w_{\ell,i} = ( \frac{\ell}{k^{2}} )^{1+\tau }w_{k^{2},i}\) and X i ≥ −b we have
$$\displaystyle\begin{array}{rcl} S_{\ell}& =& \sum _{i=1}^{k^{2}}w_{\ell,i}X_{i} +\sum _{ i=k^{2}+1}^{\ell}w_{\ell,i}X_{i} {}\\ & \geq & ( \frac{\ell} {k^{2}})^{1+\tau }S_{ k^{2}} -\sum _{i=k^{2}}^{\ell}w_{\ell,i}b {}\\ & =& (1 + o_{\tau }(1))S_{k^{2}} - o_{b,\tau }(1), {}\\ \end{array}$$from which the claim follows.
- 10.
To see this note that for m such that 2−m < c∕2, any dyadic interval of length 2−i contains a dyadic interval of length 2−(i+m) disjoint from X. Therefore, for any component any μ x, i of \(\mu \in \mathcal{ P}(X)\), we have \(H(\mu _{x,i},\mathcal{D}_{i+m}) \leq \log (2^{m} - 1)/m <1\). The porosity statements follow from this.
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Acknowledgements
I am grateful to Boris Solomyak for useful discussions, and to Ariel Rapaport and the anonymous referee for a careful reading and for many comments on a preliminary version of the paper. Part of the work on this paper was conducted during the 2016 program “Dimension and Dynamics” at ICERM. This research was supported by ERC grant 306494.
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Hochman, M. (2017). Some Problems on the Boundary of Fractal Geometry and Additive Combinatorics. In: Barral, J., Seuret, S. (eds) Recent Developments in Fractals and Related Fields. FARF3 2015. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-57805-7_7
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