Porous measures on \(\mathbb{R}^{n}\): Local structure and dimensional properties (Q2758978)

The authors show that the packing dimension of any Radon measure on \(\mathbb{R}^n\) is bounded above by a function of its (lower) porosity that asymptotically approaches \(n-1\) as the porosity approximates its maximum value \(1/2\). The result generalizes the case of doubling measures studied in [\textit{J.-P. Eckmann, E. Järvenpää} and \textit{M. Järvenpää}, Nonlinearity 13, 1-18 (2000)], where the lower porosity of a measure was introduced and shown to take values in \([0,1/2]\). The proof is obtained as a corollary of a significant result describing the structure of porous measures inside small dyadic cubes. NEWLINENEWLINENEWLINEThe dimension result is in the line of work by \textit{P. Mattila} [J. Lond. Math Soc., II. Ser. 38, No. 1, 125-132 (1988; Zbl 0618.28005)] and \textit{A. Salli} [Proc. Lond. Math. Soc., III. Ser. 62, No. 2, 353-372 (1991; Zbl 0716.28006)] connecting porosities and dimensions of sets.