Mumtaz Hussain

Mathematician

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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).

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