Measuring the uniformity of Measurable Subsets of The Unit Square

Bharath Krishnan

doi:10.20944/preprints202306.0834.v1

Preprint Concept Paper Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 9 June 2023 / Approved: 12 June 2023 / Online: 12 June 2023 (13:02:15 CEST)
Version 2 : Received: 11 July 2023 / Approved: 12 July 2023 / Online: 12 July 2023 (10:57:46 CEST)

Suppose set R is a subset of [0,1] x [0,1]. We want to define a measure of uniformity of R in the unit square using dimension d in [0,2]$ of the d-dimensional Hausdorff measure. Inorder to understand uniformity, we'll give examples in section 0 where points of R are uniform in [0,1] x [0,1]. Next in section 1, we will define preliminary definitions (e.g. Hausdorff & Counting measure) to define a uniformity of measurable subsets of the unit square. Finally, in section 2 we will define a measure of uniformity between 0 and 1 w.r.t a uniform R with Hausdorff-dimension d. (In this case, the larger the measure of uniformity, the smaller the non-uniformity w.r.t to uniform R)

Spatial; Dimensions; Uniform Distribution; Measure Theory; Hausdorff Dimension; Sparse

Computer Science and Mathematics, Mathematics

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Measuring the uniformity of Measurable Subsets of The Unit Square

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