scholarly journals Projection theorems for intermediate dimensions

2021 ◽  
Author(s):  
Stuart A. Burrell ◽  
Kenneth Falconer ◽  
Jonathan Fraser
Keyword(s):  
Projection Theorems

Projection theorems for box and packing dimensions

1996 ◽  
Vol 119 (2) ◽  
pp. 287-295 ◽  
Author(s):  
K. J. Falconer ◽  
J. D. Howroyd
Keyword(s):  
Orthogonal Projection ◽  
Packing Dimension ◽  
Analytic Subset ◽  
Box Counting ◽  
Packing Dimensions ◽  
Image Position ◽  
Almost All ◽  
Projection Theorems

AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

scholarly journals Weighted holomorphic Besov spaces on the polydisk

2011 ◽  
Vol 9 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Anahit V. Harutyunyan ◽  
Wolfgang Lusky
Keyword(s):  
Fractional Derivative ◽  
Besov Space ◽  
Besov Spaces ◽  
Regular Variation ◽  
Holomorphic Functions ◽  
Spaces Of Holomorphic Functions ◽  
Weighted Besov Spaces ◽  
Unit Polydisk ◽  
Projection Theorems ◽  
Dimensional Lebesgue Measure

This work is an introduction of weighted Besov spaces of holomorphic functions on the polydisk. LetUnbe the unit polydisk inCnandSbe the space of functions of regular variation. Let1≤p<∞,ω=(ω1,…,ωn),ωj∈S(1≤j≤n)andf∈H(Un).The functionfis said to be an element of the holomorphic Besov spaceBp(ω)if‖f‖Bp(ω)p=∫Un|Df(z)|p∏j=1nωj(1-|zj|)/(1-|zj|2)2-pdm2n(z)<+∞, wheredm2n(z)is the2n-dimensional Lebesgue measure onUnandDstands for a special fractional derivative offdefined in the paper. For example, ifn=1thenDfis the derivative of the functionzf(z).We describe the holomorphic Besov space in terms ofLp(ω)space. Moreover projection theorems and theorems of the existence of a right inverse are proved.

scholarly journals Marstrand type projection theorems for normed spaces

2019 ◽  
Vol 6 (4) ◽  
pp. 367-392
Author(s):  
Zoltán Balogh ◽  
Annina Iseli
Keyword(s):  
Normed Spaces ◽  
Projection Theorems

scholarly journals Combinatorial proofs of two theorems of Lutz and Stull

2021 ◽  
pp. 1-12
Author(s):  
TUOMAS ORPONEN
Keyword(s):  
Information Theory ◽  
Euclidean Space ◽  
Pigeonhole Principle ◽  
Algorithmic Information Theory ◽  
Orthogonal Projections ◽  
Information Theoretic ◽  
Secondary Purpose ◽  
Packing Dimensions ◽  
Algorithmic Information ◽  
Projection Theorems

Abstract Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if \[K \subset {\mathbb{R}^n}\] is any set with equal Hausdorff and packing dimensions, then \begin{equation} \[{\dim _{\text{H}}}{\pi _e}(K) = \min \{ {\dim _{\text{H}}}{\text{ }}K{\text{, 1}}\} \] \end{equation} for almost every \[e \in {S^{n - 1}}\] . Here \[{\pi _e}\] stands for orthogonal projection to span ( \[e\] ). The primary purpose of this paper is to present proofs for Lutz and Stull’s projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman’s “potential theoretic” method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to generalise Lutz and Stull’s theorems: the versions in this paper apply to orthogonal projections to m-planes in \[{\mathbb{R}^n}\] , for all \[0 < m < n\] .

scholarly journals Self-embeddings of Bedford–McMullen carpets

2017 ◽  
Vol 39 (3) ◽  
pp. 577-603 ◽  
Author(s):  
AMIR ALGOM ◽  
MICHAEL HOCHMAN
Keyword(s):  
Tangent Sets ◽  
Self Similar ◽  
Projection Theorems

Let $F\subseteq \mathbb{R}^{2}$ be a Bedford–McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between zero and one. We prove that any similarity $g$ such that $g(F)\subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by ‘zooming in’ on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self-similar sets in the line.

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