arXiv subjects

Ekta Agrawal

Publications and source records attributed to Ekta Agrawal.

3 recordsLinked to original sources

arXiv2026

Some remarks on the exponential separation and dimension preserving approximation for sets and measures

In the dimension theory of sets and measures, a recent breakthrough happened due to Hochman, who introduced the exponential separation condition (ESC) and proved the Hausdorff dimension result for invariant sets and measures generated by similarities on the real line. Following this groundbreaking work, we make a modest contribution by weakening the condition. Further, we define the modified ESC using the convex hull of the attractor and show that for homogeneous self-similar IFS on $\mathbb{R},$ both definitions coincide. We also define some sets in the class of all nonempty compact sets using the Assouad and Hausdorff dimensions and subsets of measures in the space of Borel probability measures on $\mathbb{R}^m$ using the $L^q$ dimension and the Rajchman property, and prove their density in the respective spaces.

math.DS↗

arXiv2025

Some results on Lower Assouad and quantization dimensions

In this paper, we first show that the collection of all subsets of $ \mathbb{R} $ having lower dimension $ γ\in [0,1] $ is dense in $ Π(\mathbb{R}) $, the space of compact subsets of $ \mathbb{R} $. Furthermore, we show that the set of Borel probability measures with lower dimension $ β\in [0, m] $ is dense in $ Ω(\mathbb{R}^m) $, the space of Borel probability measures on $ \mathbb{R}^m $. We also prove that the quantization and the lower dimension of a measure $ \vartheta $ coincide with those of the convolution of $ \vartheta $ with a finite combination of Dirac measures. In the end, we compute the lower dimension of the invariant measure associated with the product IFS.

math.DS↗

arXiv2025

Dimension preserving set-valued approximation and decomposition via metric sum

In the literature, the Minkowski-sum and the metric-sum of compact sets are highlighted. While the first is associative, the latter is not. But the major drawback of the Minkowski combination is that, by increasing the number of summands, this leads to convexification. The present article is uncovered in two folds: The initial segment presents a novel approach to approximate a continuous set-valued function with compact images via a fractal approach using the metric linear combination of sets. The other segment contains the dimension analysis of the distance set of graph of set-valued function and solving the celebrated distance set conjecture. In the end, a decomposition of any continuous convex compact set-valued function is exhibited that preserves the Hausdorff dimension, so this will serve as a method for dealing with complicated set-valued functions.

math.DS↗