Hostname: page-component-6766d58669-h8lrw Total loading time: 0 Render date: 2026-05-15T10:10:55.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Marstrand-type Theorems for the Counting and Mass Dimensions in ℤ

Part of: Classical measure theory Combinatorics

Published online by Cambridge University Press:  16 March 2016

DANIEL GLASSCOCK
DANIEL GLASSCOCK*
Affiliation:
Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210, USA (e-mail: glasscock.4@buckeyemail.osu.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are

$$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$
where ⌊A⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ $\mathbb{R}^d$ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings:
$$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$
where
$${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha\ \bigg| \1 \leq \|C_i\| \leq r \|C\| \biggr\}$$
in which the infimum is taken over cubic coverings {Ci } of A ∩ C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ $\mathbb{R}^d$ under orthogonal projections with range of dimension k have counting dimension at least min(k, D(A)); if we assume D(A) = D (A), then the mass dimension of A under the typical orthogonal projection is equal to min(k, D(A)). This work extends recent work of Y. Lima and C. G. Moreira.

MSC classification

Primary: 05A99: None of the above, but in this section
Secondary: 28A80: Fractals

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 5 , September 2016 , pp. 700 - 743
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Barlow, M. T. and Taylor, S. J. (1989) Fractional dimension of sets in discrete spaces. J. Phys. A 22 2621–2628.Google Scholar
[2] Barlow, M. T. and Taylor, S. J. (1992) Defining fractal subsets of $\mathbb{Z}^d$ . Proc. London Math. Soc. (3) 64 125–152.Google Scholar
[3] Bergelson, V. (1996) Ergodic Ramsey theory: An update. In Ergodic Theory of $\mathbb{Z}^d$ -Actions: Warwick, 1993–1994, Vol. 228 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 1–61.Google Scholar
[4] Furstenberg, H. (1981) Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press.Google Scholar
[5] de la Harpe, P. (2000) Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press.Google Scholar
[6] Iosevich, A., Rudnev, M. and Uriarte-Tuero, I. (2007) Theory of dimension for large discrete sets and applications. arXiv:0707.1322 Google Scholar
[7] Lee, J. M. (2013) Introduction to Smooth Manifolds, second edition, Vol. 218 of Graduate Texts in Mathematics, Springer.Google Scholar
[8] Lima, Y. and Moreira, C. G. (2014) A Marstrand theorem for subsets of integers. Combin. Probab. Comput. 23 116–134.CrossRefGoogle Scholar
[9] Marstrand, J. M. (1954) Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3) 4 257–302.Google Scholar
[10] Mattila, P. (1975) Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Ser. A I Math. 1 227–244.CrossRefGoogle Scholar
[11] Mattila, P. (1995) Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
[12] Naudts, J. (1988) Dimension of discrete fractal spaces. J. Phys. A 21 447–452.CrossRefGoogle Scholar
[13] Naudts, J. (1989) Reply to a paper by M. T. Barlow and S. J. Taylor. J. Phys. A 22 2627–2628.Google Scholar
[14] Peres, Y. and Schlag, W. (2000) Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102 193–251.CrossRefGoogle Scholar
[15] Tao, T. and Vu, V. H. (2010) Additive Combinatorics, Vol. 105 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar