A general principle for Hausdorff measure

We introduce a general principle for studying the Hausdorff measure of limsup sets. A consequence of this principle is the well-known Mass Transference Principle of Beresnevich and Velani (2006). https://arxiv.org/pdf/1808.02135.pdf

The main result is as follows.

\begin{theorem}[Hussain-Simmons, 2018]\label{HS:thm}
Fix $\delta > 0$ , let $(B_i)_i$ be a sequence of open sets in an Ahlfors $\delta$ -regular metric space $X$ , and let $f$ be a dimension function such that

(1) $\begin{align*} r \mapsto &r^{-\delta} f(r) \text{ is decreasing, and}\\ \label{f2} &r^{-\delta} f(r) \to \infty \text{ as }r \to 0. \end{align*}$

Fix $C > 0$ , and suppose that the following hypothesis holds:
\begin{itemize}
\item[(*)] For every ball $B_0 \subset X$ and for every $N\in\mathbb N$ , there exists a probability measure $\mu = \mu(B_0,N)$ with $\Supp(\mu) \subset \bigcup_{i\geq N} B_i\cap B_0$ , such that for every ball $B = B(x,\rho) \subset X$ , we have

(2) $\begin{equation*} \mu(B) \ll \max\left(\left(\frac{\rho}{\diam B_0}\right)^\delta,\frac{f(\rho)}{C}\right). \end{equation*}$

\end{itemize}
Then for every ball $B_0$ ,

$\HH^f\left(B_0 \cap \limsup_{i\to\infty} B_i\right) \gtrsim C.$

In particular, if the hypothesis \text{(*)} holds for all $C$ , then

$\HH^f\left(B_0\cap \limsup_{i\to\infty} B_i\right) = \infty.$

\end{theorem}

The condition \eqref{f2} is a natural condition which implies that $\HH^f(B)=\infty$ . A consequence of this theorem is the celebrated Mass Transference Principle of Beresnevich-Velani (2006).

Dr Mumtaz Hussain

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A general principle for Hausdorff measure

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