Abstract
We use Kolmogorov complexity methods to give a lower bound on the effective Hausdorff dimension of the point \((x,ax+b)\), given real numbers a, b, and x. We apply our main theorem to a problem in fractal geometry, giving an improved lower bound on the (classical) Hausdorff dimension of generalized sets of Furstenberg type.
N. Lutz—Research supported in part by National Science Foundation Grant 1445755.
D.M. Stull—Research supported in part by National Science Foundation Grants 1247051 and 1545028.
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Notes
- 1.
- 2.
As a matter of notational convenience, if we are given a nonintegral positive real as a precision parameter, we will always round up to the next integer. For example, \(K_{r}(x)\) denotes \(K_{\lceil r\rceil }(x)\) whenever \(r\in (0,\infty )\).
- 3.
Regarding asymptotic notation, we will treat dimensions of Euclidean spaces (i.e., m and n) as constant throughout this work but make other dependencies explicit, either as subscripts or in the text.
- 4.
Our main theorem also provides yet another alternative proof that every Kakeya set \(E\subseteq \mathbb {R}^2\) has \(\dim _H(E)=2\). Briefly, let A be the minimizing oracle for E from Theorem 10, and let \(a,b,x\in \mathbb {R}\) satisfy \(\dim ^A(a)=1\), \(\dim ^{A,a,b}(x)=1\), and \((x,ax+b)\in E\). Then Theorem 1 gives \(\dim _H(E)\ge \dim ^A(x,ax+b)\ge \dim ^A(x|a,b)+\min \{\dim ^A(a,b),\dim ^{A,a,b}(x)\}\ge 2\).
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Lutz, N., Stull, D.M. (2017). Bounding the Dimension of Points on a Line. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_31
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