\subsection{Proof of Theorem~\ref{thrm: Hdim and non-divergence}}
	
    Having established Proposition~\ref{propn: main covering propn}, the proof of Theorem~\ref{thrm: Hdim and non-divergence} follows the same lines as in~\cite{KKLM-SingSystems}.
    Let $x\in X$ and let $Z_x \subseteq [-1,1]$ denote the set of points $s$ for which the trajectory $g_t u(\vp(s))x$ diverges on average.
	To prove part $(1)$ of the Theorem, we first note that for all compact sets $Q\subset X$ and for all $N\in \N$ and $\d>0$,
        \begin{align} \label{Z contained in liminf set}
        	Z_x \subseteq \liminf_{M\r\infty} Z_x(Q,N,t,\d)
        		= \bigcup_{N_0\geq 1} \bigcap_{N\geq N_0} Z_x(Q,N,t,\d)
        \end{align}
        where the sets $Z_x(Q,N,t,\d)$ were defined in~\eqref{defn: Z_x(M, N,t, delta)}.
        We wish to apply Proposition~\ref{propn: main covering propn} by taking $Q = X_{\leq M}$ for an appropriate $M$.
        
        Fix some $t>0$ and let $M_1 = M_1(t,x) >0$ be as in Proposition~\ref{propn: main covering propn}. 
        Suppose $M>M_1$ and $\d \in (0,1)$. Then, Proposition~\ref{propn: main covering propn} says that we can cover $Z_x(X_{\leq M} ,N,t,\d)$ by at most $ C_3^{N} e^{(1-\d\b) \a(Nt) } $ intervals of radius $e^{-\a(Nt)}$, where $C_3 \geq 1$ is a constant which is independent of $x, t$ and $N$.
        
         Then, we have
        \[
        	\overline{dim}_{box}\left(\bigcap_{N\geq N_0} Z_x(Q,N,t,\d) \right)
            \leq \lim_{N\r\infty} \frac{N\log(C_3)+(1-\d\b) \a(Nt)}{ \a(Nt)} 
            = \frac{\log(C_3)+(1-\d\b) \a(t)}{ \a(t)} 
        \]
        where for a set $A\subseteq [-1,1]$, $\overline{dim}_{box}(A)$ denotes its upper box dimension.
        
        Since $Z_x$ is contained in countably many such sets by~\eqref{Z contained in liminf set} and since the upper box dimension dominates the Hausdorff dimension (which is stable under countable unions), we get that 
        \[ dim_H(Z_x) \leqslant \frac{\log(C_3)}{ \a(t)} + 1-\d\b \]
        where $dim_H$ denotes the Hausdorff dimension.
        Taking the limit as $t\r\infty$ and $\d \r 1$, we obtain the desired dimension bound.
        
        Part $(2)$ of Theorem~\ref{thrm: Hdim and non-divergence} follows from Proposition~\ref{propn: main covering propn} and the Borel-Cantelli Lemma.
        More precisely, it follows from the statement of the Proposition that the set $Z_x(M,N,t,\d)$ has measure at most $C_3^N e^{-\d\b\a(Nt)}$.
        Choosing $t >0$ (and hence $M$) to be large enough, depending on $\d$ and $C_3$, we see that that the measures of these sets are summable in $N$.
{"config": "arxiv", "file": "1806.06832/ExpRecurrence/Thrmproof.tex"}