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Article
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Summary: This article discusses the dimension theory of a family of functions' graphs, including the well-known 'popcorn function' and its higher-dimensional analogues. The box and Assouad dimensions, as well as the intermediate dimensions, which interpolate between Hausdorff and box dimensions, are calculated. Various tools, such as the Chung-Erdos inequality, higher-dimensional Duffin-Schaeffer type estimates, and Euler's totient function bounds, are used in the proofs. Applications include obtaining bounds on the box dimension of fractional Brownian images of the graphs and the Holder distortion between different graphs.
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Statistics & Probability
Lara Daw et al.
Summary: This paper investigates the image, level, and sojourn time sets associated with sample paths of the Rosenblatt process. It provides results on Hausdorff dimensions (both classical and macroscopic), packing and intermediate dimensions, as well as logarithmic and pixel densities. Additionally, the paper establishes the time inversion property of the Rosenblatt process and discusses technical points concerning the distribution of Z.
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Statistics & Probability
Stuart A. Burrell
Summary: This paper explores intermediate dimensions by using potential-theoretic methods to derive dimension bounds for images of sets under Holder maps and stochastic processes. It applies these methods to compute dimension values for Borel sets under fractional Brownian motion, establishing continuity of dimension profiles and providing insights into how Hausdorff dimensions influence typical box dimensions of Holder images. The study proposes a general strategy for analyzing dimensional information from specific fractional Brownian images of sets and obtains bounds on the Hausdorff dimension of exceptional sets in projection settings.
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Summary: In this paper, the almost sure values of the Phi-dimensions of random measures supported on random Moran sets with a homogeneity property and uniform separation condition are determined. The Phi-dimensions are intermediate Assouad-like dimensions, with their values depending on the size of Phi.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
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Article
Statistics & Probability
Kenneth J. Falconer
Summary: This research demonstrates that the almost sure theta-intermediate dimension of the image of the set F-p under index -h fractional Brownian motion is theta/ph+theta, a value smaller than that obtained by directly applying the Holder bound for fractional Brownian motion, thus establishing the box-counting dimension of these images.
STATISTICS & PROBABILITY LETTERS
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Article
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S. A. Burrell et al.
Summary: This paper investigates the fractal aspects of elliptical polynomial spirals, providing a dimensional analysis and discovering two phase transitions within the Assouad spectrum. It then uses this dimensional information to obtain bounds for the Holder regularity of maps that can deform one spiral into another, utilizing fractional Brownian motion and dimension profiles to constrain the Holder exponents.
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Article
Mathematics
Istvan Kolossvary
Summary: The intermediate dimensions of a set Lambda, denoted by dim(theta) Lambda, interpolate between its Hausdorff and box dimensions using the parameter theta is an element of [0, 1]. For a Bedford-McMullen carpet Lambda with distinct Hausdorff and box dimensions, it is shown that dim(theta) Lambda is strictly less than the box dimension of Lambda for every theta < 1. Moreover, the derivative of the upper bound is strictly positive at theta = 1. However, finding a precise formula for dim(theta) Lambda remains a challenging problem.
JOURNAL OF FRACTAL GEOMETRY
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Article
Mathematics
Amlan Banaji et al.
Summary: The study proves the existence of a family of dimensions, known as intermediate dimensions, which lie between the Hausdorff dimension and the box dimension. It also provides a necessary and sufficient condition for a given function to be realized as the intermediate dimensions of a bounded subset.
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Article
Mathematics, Applied
Jonathan M. Fraser
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Article
Mathematics
Ignacio Garcia et al.
Summary: We investigate a class of dimensions that fall between the box and Assouad dimensions, including quasi-Assouad dimensions and the theta-Assouad spectrum. These dimensions offer refined geometric information by varying in the depth of scale considered. Our focus is on intermediate dimensions complementing the familiar ones, and we explore relationships between them and other dimensions.
JOURNAL OF FRACTAL GEOMETRY
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Mathematics
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