On the mean $\Psi$-intermediate dimensions

TL;DR

This paper introduces the mean $ ext{ extPsi}$-intermediate dimension, bridging mean Hausdorff and metric mean dimensions, and explores its properties, principles, and applications with illustrative examples.

Contribution

It defines the mean $ ext{ extPsi}$-intermediate dimension and establishes its fundamental properties, relations, and practical examples, advancing the understanding of fractal dimensions.

Findings

The mean $ ext{ extPsi}$-intermediate dimension lies between mean Hausdorff and metric mean dimensions.

Established the mass distribution principle and Frostman-type lemma for this new dimension.

Provided examples demonstrating the application of the mean $ ext{ extPsi}$-intermediate dimension.

Abstract

In this paper, we introduce the mean $\Psi$-intermediate dimension which has a value between the mean Hausdorff dimension and the metric mean dimension, and prove the equivalent definition of the mean Hausdorff dimension and the metric mean dimension. Furthermore, we delve into the core properties of the mean $\Psi$-intermediate dimensions. Additionally, we establish the mass distribution principle, a Frostman-type lemma, H\"older distortion, and derive the corresponding product formula. Finally, we provide illustrative examples of the mean $\Psi$-intermediate dimension, demonstrating its practical applications.